I'll start with the text adventure language Hugo because it is one of the easier languages to deal with. Hugo game files end in .hex. Some Hugo games have graphics and sound, which come in separate files. Those files don't have any set extension; however, in most cases the graphics and sound files are bundled with the .hex file in a zipped archive. You can download an interpreter for your system (which is called the engine) from the Hugo home page. Versions of the interpreter are available for Windows, Mac, Linux, Dos, and more. You can find Hugo games in the /if-archive/games/hugo/ directory of the IF Archive.
Alan game files come in two pieces: an .acd file and a .dat file. Because of this, Alan games are usually put into a zip file so you don't have to download them separately. You can get the interpreter (which is called the runtime) either from the Alan home page or from the /if-archive/programming/alan/executables directory of the IF Archive. You can download Alan games from the /if-archive/games/alan directory of the IF Archive.
Before I can talk about TADS, I need to explain the two variants of TADS: regular, and HTML TADS. HTML TADS is a variant of TADS which allows games to have graphics and sound in them. Because of this, most platforms have two possible interpreters, a regular TADS interpreter, and a multimedia interpreter. Either version will play any of the TADS games, but since multimedia TADS interpreters can also display graphics and play sounds, I recommend that you use one of them.
TADS game files end in .gam. The games which have graphics and sound sometimes have additional resource files, which end in .rs0, .rs1, and so on. You can get a TADS interpreter from this page. Note that TADS interpreters tend to have different names on different platforms: HTML TADS under Windows, HyperTADS on the Mac, etc. You can find TADS games in the /if-archive/games/tads directory of the IF Archive.