Gaming audio has evolved a great deal from the first machines to feature it in the 1970s.
In the beginning: Beeps and clicksOn early machines, there either wasn't any sound hardware, or the sound hardware was extremely simple, being able to click a speaker or play simple tones. This was also the default sound system on IBM PC compatibles for many years, as well as on early Apple computers like the Apple II, IIe and IIc (The Apple IIGS had PCM sample playback capabilities courtesy of an Esoniq PCM Codec).
Programmable sound generatorsOne step up from simple beeps and clicks was the programmable sound generator (PSG), a set of oscillators on a chip that could be programmed in real time. The simplest ones, like the Texas Instruments 76496 and GI/Microchip 8910, had 3 square-wave channels and 1 white noise channel, all with independent volume controls. At the other end of the spectrum was the Commodore 64's SID, a full 3-channel hybrid analog synthesizer with sine, square and triangle wave oscillators, filters and white noise. The Atari ST also used a PSG audio chip, though later versions also included a PCM audio codec for rudimentary speech and sound effects support. Many PSGs could be fooled into playing back sampled audio by feeding PCM values into the volume control registers thousands of times a second, as heard on some Game Gear games by "Say-Gah!".
FM synthesizersThe next step up would be FM synthesizers. FM synthesizers work by combining tones of various frequencies together in real time, with up to 4 oscillators working together to make a note. The technique works best for percussion and woodwind instruments, and tends to sound "plasticky" or "metallic" if asked to recreate a whole orchestra. It's excellent for making strange noises and sound effects. When FM synthesis was popular, Yamaha owned the patents, so pretty much all arcade and console games that used FM used a Yamaha FM synthesizer chip to do the work. The Sega Genesis had a Yamaha FM+ TI PSG chip inside. Additionally, the Yamaha OPL chips were also found on the Mockingboard series of sound cards for the Apple II, the MoonSound, MSX Music and MSX Sound expansion cards for the MSX, and the popular AdLib sound card for PCs (and became the defacto standard until usurped by the SoundBlaster in the early 90s), as well as in most SoundBlaster PC sound cards and clones to provide AdLib compatibility.
Sample playback and PCM synthesisThe crown of gaming audio, Pulse-Code Modulation systems work from actual samples of instruments, making their sound much richer. Since they can reproduce practically any sound, all sorts of odd effects are possible. PCM engines typically don't do any mathematical synthesis on their own, preferring instead to mix samples together at various speeds and volumes; however, high-end samplers used in music composition can filter the sound and do all sorts of other tricks. DSPs may be present to add effects like echo and reverb. The first popular gaming platforms to use a PCM Synthesis were the Amiga, SNES, and believe it or not, Pinball systems, mainly those that used Midway's DCS PCM Synthesis board (which also saw use in Mortal Kombat and Revolution X cabinets, since it not only reproduces instruments more faithfully, but one of the many tricks PCM synthesis could do was transparently loop fully-voiced music tracks, which is an important feature of the latter game). The NES and Sega Genesis both had rudimentary PCM support, but this was mainly used for pre-recorded voices, sound effects, and drums, (and in actuality worked by manipulating the PSG in the case of the NES. The modified OPL2 chip (called an OPN2) used by the Genesis has a PCM codec mode, but the Genesis can also resort to manipulating the PSG to play back PCM sounds if needed). Pretty much every system introduced since uses PCM. On PCs, PCM started to take over from FM-only cards in the early 1990s, when the first sound cards with samplers on-board and audio codecs appeared. On the low end were "hybrids" which used PCM sample playback for sound effects and speech, but FM synthesis for music. Most SoundBlaster cards except the AWE, Live!, Audigy and X-Fi series of cards were these, as were the numerous SoundBlaster "clones". On the higher end we have cards like the Gravis UltraSound, Trident 4DWave, and the SoundBlaster AWE, Live!, Audigy and X-Fi series which were full PCM Synthesis devices. These used audio samples provided by the user (Patches in the UltraSound and 4DWave, and SoundFonts on the AWE, Live!, Audigy and X-Fi) for music synthesis, but offered rudimentary PCM sample playback on a separate codec for sound effects and speech as well. However on the PC end, developers were slow to take advantage of these cards and full support only appeared in the mid-90s despite the first of them appearing as early as 1991, and thus many early game ports had inferior audio. As CPU power increased, especially after the Pentium and PowerPC processors became popular (around 1995), PC games began using software PCM engines to play instruments and sound effects.
Red-Book CD audioFinally, once games started to ship on optical media, prerecorded audio became common. The audio could be played from the CD just like any other song, while the game data lived in memory. This technology actually developed in tandem with PCM Sample playback and competed with PCM Synthesis, and is sometimes used together with the former (for example, in the PC port of Wipe Out and Quake II, where the music is played from the music CD partition of the disc while the sound effects are played through PCM sample playback). A nice side-effect of this would be that the game CD is it's own soundtrack CD and the soundtrack can be enjoyed on any regular CD player, and it also adds an extra layer of complexity for Copy Protection in that multi-partition game discs are difficult to duplicate reliably. Additionally, the music sounds just as good as PCM sampled music. On the downside, however, looping music tends to be difficult if not impossible to implement- as evident in Sonic CD on the Sega CD, where the music had a short fade-out and fade-in section when repeating. This was one of the main draws of the Apple Macintosh in the early 90s, when educational and adventure games alike started using these for music as an alternative to FM synthesis. See also Video Games, Pac Man Fever.