Many subsets of Fourier transforms are useful in compression. For example, JPEG, MPEG-1 and MPEG-2*, MJPEG, and many other CODECs work based on DCTs, which are a simplified subset of FFTs. (MPEG-4 also supports all of MPEG-1 and MPEG-2, but primarily uses a bunch of other compression mechanisms for its various layers, mostly wavelet and procedural, rather than DCTs.) Basically only the most dominant waves are stored, and their characteristics are compressed using Huffman or similar. This technique is known as lossy compression, since only the most dominant details are preserved (which are usually the most important, but this leads to problems with medical images being transmitted by people who don't understand that JPEG loses a LOT of fine detail).

* Contrary to popular belief, MP3 is not MPEG-3 (which doesn't exist); it's part of MPEG-1. MP3 gets its name from its common file extension, which is derived from it being MPEG-1 audio layer 3. Many low-bitrate so-called-MP3s are also actually compressed using layer 2. Unfortunately, when people see an MPG file they think movie, just like how when they see a WAV they think sound, even though both formats are multi-chunk, multi-layer, and multi-talented. In fact, the reference implementation of the MPEG-1 Layer 3 CODEC puts the audio chunk into a WAV file, rather than an MPEG file, because that's a more proper way to do it (and yet, most programs out there are lazy, as are most people when it comes to understanding these things, but the whole point to the WAV and MPG et al formats is so that people don't have to know the difference, since it's supposed to be up to the programs to grok this). (This has been a MaggieRant(tm). Consider yourself better-informed for digging around in completely-unrelated topics. :)