Klingon, Dothraki, Na’vi, Sindarin. These are fictional but fully fleshed-out languages, also known as conlangs (constructed languages). There are many others as well, created for practical or artistic use.
Conlanging is not just for fantasy writers or professional linguists; it can be a fascinating pastime for anyone with a passion for linguistics, creative writing, or language in general. Learn more about how you can construct a new language of your own and how this process can affect your understanding of linguistics and the creative writing process.
For a more detailed look on how to create a conlang, check out this post by Nick Totten.