Chiptunes are pieces of music composed for the sound chips or other archaic output devices of vintage computers. The point being that you're severely limited in terms of the hardware's capabilities so you have to think outside the box to make it do what you want, and the result is unlikely to sound like any real instrument. On the other hand the programming capabilities are enormous (assuming a decent tracker exists for the hardware you're working on) and that allows for some incredibly rich content.
Probably the most famous example of a sound chip being taken seriously by musicians is the MOS SID, the sound chip from the Commodore 64. There are lots of other platforms that are popular amongst chiptune artists though, such as the Nintendo Game Boy (aka DMG if you're taking things seriously), Nintendo consoles, Yamaha OPL chips (FM synths as found on 16-bit-era sound cards) and many computers that didn't really have sound chips at all but have some digital output that can be coaxed into making sound, such as the Sinclair Spectrum.
I'd like to link to some of my favourite pieces of music but sadly most the best stuff I've found has been via 8bitcollective, which seems to have been erased from the face of the internet (I suspect foul play by RIAA lawyers :o( ). Here's some links to sites where you can find some stuff to listen to though:
And finally, if you look in the player on the bottom left, you'll find one of my efforts at creating chip music, Eighty Watt Electric Gecko - made using a Yamaha OPL3 and Adlib Tracker II.