REX files and ACID files, in the simplest terms, are audio loops. They have some extra formatting that makes them REX or ACID files, though.
REX files are created using a program from Propellerheads called ReCycle. That’s how it gets it’s name; REX = REcycle eXport. The ReCycle program is dedicated to creating REX files for use in other programs, such as Reason’s REX player, or directly in an audio program that is capable of utilizing this functionality, such as the current version of Pro Tools.
ACID files are basically just WAV files that contain “extra” information about the WAV file. Information such as tempo, length, and a beatmap are usually included in the “extra” information. Some ACID files (especially older ones) exclude the beatmap information which makes them useful in other programs (like Pro Tools), and only include tempo and length information. ACID files are created using a program from Sony called, well, ACID. ACID can be used to create acid files, but the program itself is also a detailed and complex looping DAW of it’s own. There are other programs that can be used to create ACID files, such as Sony’s Sound Forge, and a few other third party programs.
What these programs do is turn a regular audio loop into a REX/ACID file by separating the loop into smaller pieces, usually on the individual beats within the loop. This process is called “slicing” the loop. REX files and most ACID files can be and are considered “sliced” loops.
REX/ACID file usage all depends on the functionality of the program you are using them in. Some programs, such as Reason’s REX player, allow you to trigger each slice in the files with a MIDI event.
Some programs (such as ACID and Ableton Live) do not require the slice data, and only go off the tempo and length information to “warp” the file to conform to a specific tempo.
Some programs, such as Pro Tools, allow you to use the file in a tick based audio track, where the slices lock to the beats in the track, and follow the tempo of the song if/when changed. Programs like this also allow you to cut/copy/paste individual slices anywhere you like.