The question of just how many languages Tolkien constructed is much more difficult to answer than a naive questioner may realize. It all comes down to definitions. By the strictest definition, Tolkien didn't make even one single language. By the most liberal definition, he made a virtually indefinite number of languages.
We will start with the assumption that if you devise words and grammar, the result CAN be termed a language; it is not merely some kind of queer "literary art" that merely happens to imitate the structure of "real" languages. Of course, if you insist that a system of words and grammar is not an actual language unless it emerged and evolved more or less spontaneously in a community of people who spoke this language for generations, then Tolkien didn't make any languages at all. After all, Tolkien's linguistic constructions have never been anyone's mother tongue.
The answer to the "How many languages?" question will likewise be zero if you mean languages that are so "complete" that you can readily translate any text into them. Tolkien didn't develop an Elvish terminology that can be used to discuss brain surgery or quantum physics. Indeed the published vocabulary of even the most highly developed Elvish languages is pretty basic, and often even less than basic. Material that is still unpublished will certainly fill some of the present gaps if this material is eventually made available, but we should not think that any Tolkien-language was ever even remotely "complete" in terms of vocabulary. (On the other hand, Tolkien has left us so much material that with some ingenuity, one could develop a much fuller vocabulary by starting from Tolkien's roots and applying his methods of derivation. Some would like to do this, and preliminary projects are indeed on their way. Others are distinctly unenthusiastic about such "fabrication" and feel that Tolkien-linguistics proper should only focus on Tolkien's own material; those who hold this view are not particularly interested in "using" the languages at all.)
When trying to count Tolkien's languages, one complicating factor is the fact that he very frequently revised them. For instance, in the Etymologies (a very important pre-LotR source document), there are examples indicating that in the late thirties, Tolkien used -n as the Quenya genitive ending. However, while writing LotR, he revised this, deciding that the Quenya genitive ending should be -o instead. Does this imply that the Quenya of the Etymologies is not really the same language as the Quenya exemplified in LotR? Must they be counted as two different languages? If a new language is born every time Tolkien made some major or minor revision, then he is the maker of hundreds or even thousands of languages.
Tolkien's endless revisions also complicate the "how many languages?" question in another way. The earliest forms of his Elvish languages were in effect revised out of existence as the decades passed by. For instance, the earliest conceptual predecessor of the Celtic-sounding Elvish language Sindarin was called Gnomish -- a highly developed language with thousands of words, set down in a dictionary written about 1915. However, so many revisions separate Gnomish and Sindarin that a speaker of the latter would never have understood what a speaker of Gnomish was saying. The two languages have relatively few vocabulary items in common, and important grammatical features (such as plural formation) are also quite different. Yet "Gnomish is Sindarin" (as Christopher Tolkien put it) in the special sense that Gnomish is the language that eventually evolved into LotR-style Sindarin in Tolkien's notes. So do we count Gnomish and Sindarin as one language, or as two? There are at least three possible angles here, each of them perfectly valid and defensible:
1) We only consider the languages that form part of the more-or-less "final" form of Tolkien's mythos, the languages known to the people of Middle-earth; in this case we would count Sindarin as one language and ignore Gnomish altogether. The latter tongue, as an actual body of words and grammar, was never at any point spoken in the universe of Frodo Baggins.
2) We are interested in the languages Tolkien actually made, irrespective of the fictional context (and irrespective of whether Tolkien "rejected" something or not; his revisions only mean that we have more ideas to study). If so, Gnomish and Sindarin are so different that they must clearly count as two entirely distinct languages.
3) We are still only interested in the languages Tolkien actually made and still do not consider the fictional context, but our study includes the actual history of development of the individual languages, in which case the Gnomish material is logically included in our study of Sindarin: We are not content with learning about LotR-style Sindarin, we also want to know the real history of this language -- how Tolkien developed it over the decades. Logically, we then consider Gnomish, Sindarin and all the intermediate stages (such as the "Noldorin" of the Etymologies) one single language.
In addition to the actual, "external" evolution of these languages in Tolkien's notes, there is also the fictional, "internal" history to consider. A highly important aspect of Tolkien's language construction was the outlining of the development of the languages over vast periods of time; the grand vision of an evolving language family was probably more important to him than displaying in great detail the "classical" forms of these tongues. In Tolkien's notes, we have "Primitive Quendian" as the supposed ultra-primitive tongue underlying all later Elvish languages. Further down the imagined time-line we have such shadowy entities as "Common Eldarin", "Pre-Record Quenya", "Pre-Historic Sindarin" etc. When explaining the origins of certain Quenya or Sindarin words, Tolkien often cited forms belonging to these supposed primitive languages. For instance, he traced Quenya alda and Sindarin galadh, both meaning "tree", to a common original *galadâ (the asterisk before the latter form indicating that it is "reconstructed" and "unattested"!) When attempting to count how many languages Tolkien made, do we include the "reconstructed", primitive tongues supposed to underlie these languages? Tolkien cited so many "primitive" words that pre-historic Elvish has more actual substance than some of the languages of the "historical" period (e.g. the very poorly attested Nandorin or "Wood-elven" tongue).
And of course, there is the extra problem that Primitive Elvish was no more immune to Tolkien's revisions than were the "historical" forms of the Elvish languages. Tolkien's early "Lexicons" (1915-17) presuppose a vision of Primitive Elvish that in some respects differs from Tolkien's later ideas. Are we dealing with one or several primitive languages here? How do we count them?
Then there are a few languages that have no direct connection to the Middle-earth mythos, predating even the earliest forms of Tolkien's narratives. In his youth, Tolkien and his friends played with nonsense-languages like "Animalic" and "Nevbosh"; later Tolkien made a private language called "Naffarin", and he also tried to extrapolate new Gothic words to supplement the known corpus of this poorly attested Germanic language. Do we include Nevbosh, Naffarin and Neo-Gothic when trying to count the languages Tolkien constructed?
If we limit the scope to the languages belonging to Tolkien's Arda mythos, and do not consider either "conceptual predecessors" or the supposedly "reconstructed" primitive forms, a useful answer to the question "How many languages did Tolkien make?" would have to go something like this:
Two of his languages -- Quenya and Sindarin, the latter incorporating the "Noldorin" material -- are relatively highly developed with thousands of words and comprehensive grammars (though only a fraction of the grammatical writings has been published). These two are the only Tolkien-languages that come close to be "useable" -- in the sense that you can with some ease write long texts in these languages if you deliberately avoid or work around the gaps in our knowledge. Tolkien himself left us a number of relatively substantial Quenya and Sindarin texts, mostly in verse. (The Sindarin text corpus is however much smaller than the Quenya text corpus.)
Three or four other Elvish languages, Telerin, Doriathrin/Ilkorin, and Nandorin, are primarily known in the form of vocabulary items numbering from ca. 30 and up to a few hundred (only for Telerin do we have a few short sentences of actual text). If we are dealing with the "classical" form of Tolkien's mythos, some would exclude Doriathrin/Ilkorin from the count: These were originally conceived as the languages of western Middle-earth in the First Age, but later Tolkien seems to presuppose that Sindarin was spoken in this area, the language of Doriath being simply an archaic form of Sindarin rather than a separate tongue. The Mannish language Adûnaic (Númenorean) primarily manifests as a never completed "report" of its structure and development; only a handful of sample sentences and a vocabulary of less than 200 words exist. The known vocabulary for Khuzdul, the language of the Dwarves, is likewise tiny, but Tolkien mentioned that he had sketched this language in some detail of structure (for their own reasons, the group currently editing Tolkien's linguistic material for publication will not discuss whether these notes survive). Some very sketchy notes setting out certain rudiments of Westron, the supposed "real" Common Tongue of Middle-earth (represented by English in the books), are preserved in the Tolkien Collection at Marquette; an attempt to make sense of this material appeared in Tyalië Tyelelliéva #17. Less than 200 Westron words are known, so the language is in no way useable. A grammar of a Mannish language called Taliska is also reported to exist, though it has never been published. (A few early Mannish words are mentioned in WJ:238, 270, 309; whether this material is compatible with Tolkien's writings on Taliska remains to be seen.) Including Quenya and Sindarin, this makes about ten languages that have at least a minimum of substance and structure.
Then there are the languages that are entirely fragmentary: The Black Speech, the language of Mordor, only manifests in the inscription on the Ring, in a single Orkish curse (for which Tolkien offered several contradictory translations) and as a very few isolated vocabulary items. Valarin, the language of the "Powers" or Gods, is likewise a tongue we have only had a glimpse of: Tolkien mentioned some 30 isolated words, but there is not a single actual sentence or phrase joining together several words.
Finally we have the languages that are virtually or entirely fictitious; even by a very liberal definition it hardly makes sense to say that Tolkien "constructed" these languages. The language of the Rohirrim is "represented" by Old English in Tolkien's narratives (just like Modern English represents Westron); a very few "real" Rohirric words are cited in various sources, but the published vocabulary does not even amount to ten words. The known Dunlendish vocabulary only consists of one single word, forgoil = "strawheads"; we don't even know how to break forgoil down into elements meaning "straw" and "head(s)". The Orcs are said to have used many barbaric tongues and languages among themselves, but except for a few Orc-names that hint at the general style of these tongues, virtually nothing is known about them (some originally Black Speech vocabulary items are said to be widespread, e.g. ghâsh = "fire"). The Avarin Elves in the eastern parts of Middle-earth are said to speak many different languages, but as far as we know, all the words Tolkien recorded of these languages are six Avarin cognates of the Quenya word Quendi "Elves" (in six different Avarin languages, of course -- each of these unnamed tongues thus has an attested vocabulary of precisely one word!) The "Woses" or Wild Men are said to have called themselves Drûg and the Orcs gorgûn; this is just about all that we know of their tongue. Even more poorly attested is the language of Harad: Gandalf on one occasion said his name "in the south" was Incánus; according to one source this is a word from the language of the Haradrim, meaning "North-spy". And we mustn't forget Entish: Tolkien provided some very general observations about its longwinded structure, but only a single Entish phrase with no translation is provided (and the transcription of this one phrase is said to be probably very inaccurate; we are told that Entish could hardly be reduced to writing at all).
So in summary: If we consider the "historical" versions of the tongues that are relevant for the classical form of the Arda mythos, Tolkien developed 2 languages that are vaguely "useable" (in the sense that you can compose long texts by deliberately avoiding the gaps in our knowledge), named roughly 8-10 other languages that have a minimum of actual substance but are in no way useable, provided mere fragments of at least 4 other languages, and alluded to numerous other languages that are either entirely fictitious or have a known vocabulary of only one or a very few actual words.
The short answer to the "How many languages?" question must go something like this: "Apart from the extremely fragmentary or entirely fictional ones, he provided varying amounts of information about some ten or twelve languages, but only two of them are highly developed with really substantial vocabularies."Ardalambion Index