Video game music has been around almost since video games were capable of producing sound. The earliest video game music consisted of beeps and boops, limited by the primitive sound technology of the time. The NES offered several titles with highly memorable tunes: The Legend of Zelda, Super Mario Brothers, Final Fantasy, and others. These did consist of the same old beeps and boops, but they were masterful beeps. The SNES offered much more sophisicated sound hardware, with support for various synthesized instruments. It also had the game with what is sometimes considered the finest game soundtrack ever made: Final Fantasy VI (released as number III in the US). Terra's Theme stands as the high point of this soundtrack.
The N64 didn't offer anything monumental in the way of video game music. The Playstation, on the other hand, offers games with both actual, live recordings, and some very fine synthesized pieces. Chrono Cross comes to mind as a good soundtrack of this type: its first and last tracks are actual recordings, but every other track is synthesized. Metal Gear Solid's music was entirely recorded from a live orchestra, though it offers a relatively low quantity of tracks.
PC's have had music as well. One or two of the very earliest PC games (and these are old enough that the term "PC" gets into murky waters) actually modulated the cathode-ray tube itself, such that it broadcasted music on a specific AM frequency that could be picked up on a decent AM radio. Other old games came with a casette tape of music, which was intended to be played in parallel with the game.
Some games used the PC's internal speaker for music, but these soundtracks were almost invariably grating on the ears. To make matters worse, some of these games did not offer the ability to turn off the music (further proving what I always say: the most important part of any feature is the ability to turn it off). Some of the old Sierra text adventures featured music like this. Thankfully, most of them let you turn it off with a press of the F2 key.
Later, with the advent of actual ISA sound cards (and later PCI, of course), PC game music could approach NES music at a technical level. The MIDI standard, too, played an important role in early game music (SimCity 2000 featured a suprisingly decent MIDI soundtrack). The eventual widespread adoption of CD-ROM drives on computers allowed the inclusion of Redbook Audio in games. It's about here that PC game music got many of its greatest soundtracks. Though not all of these games actually used Redbook Audio, they did take advantage of the CD's storage space for some really kickass music:
Some of these games are older than others, and got in earlier on this type of music. The only other type of music that has become prevalent for the PC is compressed audio such as the MP3 format. Warcraft III is the only example of such a game that I can think of off the top of my head, but I'm sure others do exist and will come out in the future.
The idea of having famous musicians compose the music for video games is often considered to have started with Quake, with NIN's Trent Reznor. The idea, which really seems like a good one on the surface, seldom takes root, with several exceptions (such as Ogre from Skinny Puppy doing the soundtrack to Descent 2). Usually, game music is done by composers who just do game music, or sometimes for movies and games, people such as Jeremy Soule (Total Annihilation, Icewind Dale) or Yasunori Mitsuda (Chrono Trigger, Chrono Cross).
Squaresoft deserves special mention, I think. Almost every game they have made since Final Fantasy has boasted a phenomenal soundtrack. Almost all of their major RPG's (most of the Final Fantasy series and the Chrono games) have their soundtracks available for purchase in CD form. I highly recommend (as said above) the Final Fantasy VI OSV, as well as the ones for both Chrono games. In fact, I'd recommend you get the chance to listen to any of the soundtracks I've mentioned in this writeup.