The current most common mental model for classifying conlangs is the Gnoli triangle, the idea that all fake languages are auxlangs, artlangs, or engelangs, or some combination of the three. If you substitute in Esperanto, Elvish, and Lojban you have the general idea. Each of these three languages inspired boatloads of imitators, some imitating the functional structure, some people imitated the goals and methodology. As long as people were unimaginative, this model usefully categorized the scene.
Ghostlang - Hebrew, Cornish, Living Latin, reconstructed PIE, and things like a hypothetical reconstructed Virginian Algonquin. These language either try to be ghost compatible (even the ghosts understand the modern speakers) or to simiply exist at all (which requires some sacrifice of academic and historical purity, especially for remnant languages)
Homelang - A language that was created for the purpose of speaking at home with children and spouse. Not to be confused with auxlangs because of the different scope and no particular restriction on design (not necessarily an all-loan-word design)
Zoolang- A language that was created so that it would be usable with animals or hypothetically evolved animals. The difference between this and Elivish is that in Elvish, all of the Tolkien languages are human languages with no particular inspiration from the sounds animals can create or the syntactical limitations of a hypothetical species that might evolve from a whale, walrus or ant (who communicates by laying down chemical trails, not by sound).
Xenolang- Xenolangs come in at least two varieties, languages that are closely related to the standard artlang model, these languages attempt to be non-human by violating possible human language universals. Na’vi and Klingon all take steps in this direction. The other sort of xenolang is for creating a language that can be transmitted by space probe, space ship or radio signal to aliens living on another planet in another solar system, or a first contact language.
Smalllang-(hmm, anyone want to try to create a compound word, smallllama?) These are small languages with small closed lexicons, and small closed sets of grammar rules and can be used for a variety of applications- pedagogy, robots, attempts to teach bonobos language, experimental linguistics (where you use a fake language to get people’s linguistic behavior separate from their mother tongue). It take several thousand hours to learn a language poorly, small languages let students and researchers get through the process in a much shorter time.
Medical Conlangs- Famous examples would be Blisssymbols and the languages that some deaf-blind people use. Languages optimized to prevent or address a medical condition. Sign languages already exist as natural languages, but for disabilities that go beyond that, new systems need to be devised that are both full languages, have a community, etc. The guy that had to dictate his autobiography via eye-blinks probably wishes there was a better binary conlang.