This question is still considered a major problem, a stumbling point of the Indo-European linguistics which does not allow linguists to settle a lot of issues of ancient linguistic history of the family. In order to have an opportunity to rely on solid language data, comparative linguistics needs firm proofs of kinship between different languages and language groups. If, for example, Celtic and Italic are similar somehow, we should be sure that at some stage in the deep past people spoke a Common Italo-Celtic language, and therefore further processes both groups undergoing are also similar and can be explained by comparing them.
But if languages look like each other or sound like each other, or their lexicons have the majority of common roots, that does not always mean that they used to be one some time ago. The thing is that tongues which are spoken in neighbouring areas will contact with one another and so borrow certain traits and exchange certain words and morphological forms. Pronunciation is similar, grammar is also similar, but languages belong to completely different groups or even families - all they did is communication which made them alike.
This is the problem of distinction between linguistic kinship and a language union. Such unions exist practically in every region of the world where people have to mix and contact intimately. Sometimes tongues representing completely different families collide and create a kind of an union: a typical example is the languages of the Caucasus which, once quite different from each other, turned out to be a language union by their phonetics and partly morphology. Armenian, a member of the Indo-European family, Georgian, numerous North Caucasian languages have elaborated a common articulation system which is now used in all of them. Due to this process, Armenian lost practically all its Indo-European distinguishing features in phonetics under the influence of Caucasian languages (including also Turkish and Persian).
However, the case with Caucasian languages is one of the simplest in this collection. Nowadays all peculiarities of the linguistic history of the Caucasus seem understandable. But a century and a half ago serious linguists considered Armenian a Caucasian, non-Indo-European language, basing on its phonetic structure.
The idea of linguistic unions was suggested by Russian linguist Trubetskoy, who wrote an article in 1923 named "The Tower of Babylon". Trubetskoy was the first ot introduce the very term and the first to distinguish a difference between a language family and a language union. According to Trubetskoy's definition, a language union first influences the syntax and morphology of languages which have to communicate closely, develops a similar phonetic structure, the true similarity of which is actually only in appearance. A long period of mutual contacts and influence on each other also make cultural background of languages similar. However, by Trubetskoy, languages of such a union are not connected by common sound correspondences and elementary original lexicon.
Later, the main principle of a language union was declared as "words differ, grammar alike". Theoretically, the process of acquiring a set of common linguistic traits was called convergention (from Latin convergo 'I come closer'), which can be either contact - as a result of close and lengthy communication between tongues, or substratum - as a result of aboriginal influence on both languages causing identical elements in them.
We will try to analyze two of the most well known cases which illuminate
the whole problem.
Trubetskoy developed his theory, basing on one of the typical examples of linguistic unions - the Balkan union, which consists of Romanian, Bulgarian, Serbo-Croatian (partly), Macedonian, New Greek, Albanian languages and several minor East Romance tongues like Aromanian and Megleno-Romanian. But the first who noticed the common features of these tongues was Franz Miklosic in the 19th century.
These languages all belong to the Indo-European family, but to different groups within it. Still, they are closely connected geographically and historically. This kind of alliance began to form even when Thracian and Illyrian languages were spoken on the Balkans. They both had a set of common traits in syntax and morphology with Greek dialects and other languages of the region - Messapic, Venetic, Old Macedonian. Later process of hellinization of the Balkans caused the formation of several common features in all languages which were spoken here after the fall of the Roman Empire. Some linguists believe that Greek was the primary language which influenced greatly all its neighbours and introduced the complex of new characteristics which are now called Balkanisms. One of the most sophisticated defenders of this Greek-dominated point of view was Christian Sandfeld. However, according to linguistic data, Albanian and Macedonian (Slavic) languages nowadays contain much more specific Balkanistic traits than New Greek, so the center of the union can be either Albania or Macedonia.
The Albanian language, a descendant of the former Illyrian and Thracian tongues, did not preserved much in morphology from Illyrian which was genetically closer to Italic and Celtic than to Ancient Greek. But while the Balkan union was forming, Albanian gradually lost all its Celtic-like traits, and its Illyrian origins can now be proven only with the use of lexical similarities of modern Albanian words with what was found of the Illyrian glossary. As for Bulgarian and Macedonian, they are representatives of the Southern branch of the Slavic group, and remain strongly Slavic with their lexicon and historical phonetics. But their morphology, especially that of nouns and verbs, changed so greatly since Slavs reached the Balkans in the 6th century, that nowadays they are strange to look at: none of the two, for example, preserved the basic noun declension which used to be quite rich in the Common Slavic language.
The list of similarities which compose the Balkan union includes the following (based on the 20 characteristic features of the Balkan languages emphasized by Christian Sandfeld in 1926):
1. Similar rhythmic organization of speech (rhythm and bars)
2. The espiratory stress in all languages except Serbian with its specific polytonic stress
3. Lack of quantitative difference in vowels of all languages (except a few Bulgarian and Macedonian dialects). This is interesting because Ancient Greek made strict distinction between long and short vowels, so did Thracian, as we may suppose. Greek lost this distinction in the Byzantine era, when vowels suffered serious changes, either under the influence of other Balkan languages or under its own internal process of development.
4. A special dorsal vowel sound used in Bulgarian, Albanian in Eastern Romanian dialects. This sound is also called neutral, and is always written like e with dots over it in Albanian, a with a smile over it in Romanian, and given a special sign in Bulgarian Cyrillic orthography.
5. As for morphology, most of the Balkan tongues have their dative and genitive cases coincided in one, though this process was never under way in Slavic languages (Romanian may be the language which introduced this feature).
6. The article is placed after the noun in Bulgarian, Macedonian, New Greek, Albanian: this seems really strange and unique both for Slavic and for Hellenic languages - the former never having articles before, the latter having it in Ancient Greek before the noun, as in most of the Indo-European tongues.
7. Balkan languages have successfully lost the infinitive. The Proto-Indo-European language had no infinitive, only several verbal nouns with the infinite meaning, but this is not the case with Balkan languages - they developed their infinitive forms (Slavic in -ti, Hellenic in -ein, Romance in -se / -re) but then lost it for unknown reasons. The infinitive was replaced by some subordinate clauses sentences with conjunctive principal clauses.
There is also a number of similarities in syntactic structure and in
lexicon. However, all common lexicon of the Balkan region was proved to
be composed just of loanwords: the original lexicon of Bulgarian remained
Slavic, and the words of Greek remain Hellenic. But during the mixture
of nations in the Middle Ages languages exchanged a great number of lexical
items: Greek and Romanian borrowed much from Slavic (remember that Slavs
used even to live in Greece in the 7th century - see Indo-European
Chronology), Slavic took a lot from Greek and Latin after Slavs were
converted to Christianity, and all Balkan languages suffered strong influence
of the Turkish language, for they were under the rule of the Ottoman Empire
for more than four hundred years. For example, Albanians, who also changed
their religion to Islam, accepted a full Turkish-like system of morphology.
Substratum words (i.e. words from Illyrian, Old Macedonian, Pelasgian and
Thracian origin) also make this layer of common lexicon wider.
Now let us turn to the problem of Baltic & Slavic roots - a problem which has been discussed for more than a century in Indo-European linguistics. First Baltic languages were touched in the beginning of the comparative era - in the middle of the 19th century when Rasmus Rask and Franz Bopp first attempted to write a comparative Baltic grammar. Along with Baltic studies, Slavic research was under way since the mid 19th century, when Miklosic published his comparative Slavic grammar (1852-1875).
Linguists were surprised to see how Baltic and Slavic languages are similar in lexicon, in grammar and in phonetics. Really, facts state that Slavic languages are the closest relatives of Baltic ones than any other languages in Europe. Let us look at a brief list of common forms in both groups:
a) satem reflection of Proto-Indo-European palatal consonants (though quite inconsistent, especially in Baltic)
b) transition of labiovelar consonants into velar ones (i.e. *kw, *gw, *gwh > k, g, g)
2. In morphology:
a) genitive singular ending in -ó
b) indirect cases with the suffix -m- (i.e. dative and instrumental in singular and plural; this is also true for Germanic)
c) locative plural ending in -su (the same in Germanic)
d) the ablative case coincided with the genitive
e) modal or participle forms with the suffix -l- (in Slavic - everywhere, in Baltic only Old Prussian shows this in the subjunctive mood -lai, -laimai, -laité)
f) participle medium voice present with the suffix -mo- (also found in the Anatolian branch)
3. In syntax:
a) the whole system of noun syntax and the identical use of cases; the archaic structure of noun declension, which preserved six-seven cases all of which can be used without prepositions
b) the rich system of verbal nouns (participles, semi-participles in Lithuanian, verbal adverbs, infinitives)
c) free word order without any signs of transition to a fixed one which happened in Romance, Celtic, Germanic and other IE groups
4. In lexicon: around 60% words have common roots, and not only the so-called "primitive lexicon".
The statement of close origins of Baltic and Slavic founded a new wave in comparative linguistics - the theory of the common "Balto-Slavic language" was born in Europe. According to this theory, the Proto-Indo-European dialects included the Balto-Slavic one, which existed for some time somewhere in Southern Europe before single Common Slavic (or Proto-Slavic) and Common Baltic (or Proto-Baltic) appeared within it. The breakup of the Balto-Slavic community is believed to have happened because of the geographical dispersion of the nations which spoke the Common language. By the end of the second millennium BC (about 1100 BC) Slavic and Baltic people, as we can judge from the linguistic and archaeological data, spread over the lands of modern Lithuania, Poland, Czech Republic, Belorussia, North Ukraine and maybe also the Danube valley in modern Hungary. Evidently, the language could not remain the same, as the distance between different tribes appeared too long for communication, so soon Slavic and Baltic dialects appeared.
The proofs of this theory are still numerous, and it still remains one of the favourites. Slavic and Baltic phonetic and morphological structure are amazingly similar, sometimes seem really identical, and their lexicon also preserves about 60% of common words. Both groups passed through the same stages of the language development, both remained quite archaic, and even after their separation from each other, similar processes were under way in both of them.
However, this is not the only point of view on the issue. The closeness of phonetics and morphological system could result not from some Proto-language common for Balts and Slavs, but from their hypothetical origins from the same group of dialects within the Proto-Indo-European community. Slavic and Baltic, as we can state, were the last to leave the Indo-European homeland, after the majority of other branches had already drifted apart. Within the area of the Late Proto-Indo-European speech, wherever it can be situated, Baltic and Slavic were close geographically. Then both nations - together - left the lands in the south and went north to the territory where they exist today. This lengthy contact caused both languages' identical development and similar structure in general.
Still another strong version exists - it claims that intimate contacts between Common Baltic and Common Slavic languages took place already when they reached Eastern regions of Europe and settled down between the Vistula and the Dvina (Daugava) rivers. Two parallel processes were going in this period between the languages - 1) people which spoke them mixed in the forest zone of Eastern Europe, where tribes could not live densely, and 2) they were influenced by substratum speech of aboriginal nations of the area. This is partly witnessed by archaeological research: Slavic and Baltic settlements in the 1st millennium BC and the 1st millennium AD were mixed in modern Belorussia, Poland and Eastern Russia. We need to stress that two thousand years ago Baltic tribes inhabited a wider region than today - their traces can be found along the Dniepr river in the Ukraine, in Belorussia and in European Russia. Moreover, close ties existed between Baltic and Thracian languages which is proven by common lexical data (see Thracian dictionary). In the 10th century Baltic population still existed in the Protva basin, in Moscow region, where Russian manuscripts place the Galindan language. Nowadays a lot of scholars of authority support the idea of "mutual penetration" of Baltic and Slavic languages into each other, though they used to be independent from each other before.
It is widely known that the lexicon of both groups includes plenty of common words, many of them were borrowed from Slavic into Baltic or vice versa. But still, linguists fail to prove that it happened in the pre-historic period: more likely, this number of words appeared in the language of Baltic tribes when Eastern Slavs already established the state - such words are mainly "civilized" (like Russian kniga - Lithuanian knyga 'a book', Russian rubez' - Latvian robez' 'a border', Old Russian gramota 'a manuscript'- Latvian gramata 'a book').
In other words, the problem of relationship between Slavic and Baltic languages is more a question which we formulated as the title of the present essay: language kinship or language union? The list of theories elaborated on this issue is not over: another version is that the Slavic language used to be a dialect of Baltic and detached from it rather late. This means, that Slavic was not one of the Indo-European branch, but a late offspring from the Baltic group, which developed its own [peculiarities on the southern edge of the Baltic area due to contacts with Iranian and other steppe tribes.
I would not decide which of the above theories is right. Each of them has its supporters, each has rather strong arguments in its favour, but none is still a winner. As everywhere on this website, I would like to avoid statements and claims - I just would like the audience to understand the point of the problem, and then to try to research the true clue to it. Our materials (for example, Lithuanian and Common Slavic grammars) can be of some help to you in this research.
Definitions of language unions seem very clumsy, and every particular case requires a special approach in order to decide whether two languages are relatives or just long-time neighbours.