One look, and you'll be able to tell why aquarium hobbyists almost all aspire to keep this fish--the beauty and elegance of the discus is unmatched among tropical FW fish. These gentle giants glide serenely through the water and are usually the picture of tranquility; the one shown above is "Big Blue" and is about 6 or 7 inches. There are quite a few species and subspecies/breeds of discus, with new ones coming up all the time in all colors of the rainbow. Their colors can show their moods; having stripes, as Big Blue shows above does indicate to some extent his stress level (he was scared by a recent water change when that pic was taken).
Discus are not considered easy fish to keep, and so I would discourage any beginner from getting them as their first fish. They require a lot of maintenace; they tend to like water which is on the soft (pH 5.5-6.8) and hot side (above 80 F); some people keep their discus temperatures as high as 86 F, which is too hot for many tropical fish). They are also picky about having extremely clean water--so that means lots of good filtration, vaccuuming up uneaten food, and water changes, water changes, water changes.
They can be sensitive even to impurities in tap water that other fish don't notice, so some people choose to use reverse osmosis to produce their water, afterwards treated with something like "Electroright" to make up for the trace elements they do need.
When not kept in conditions which are to their liking, they become more susceptible to diseases like HITH and will refuse to eat. Most discus are somewhat choosy eaters to begin with, and some will not take flake foods. They should be fed on a staple diet of things like frozen brine shrimp, bloodworms, other supplements; some people feed beef heart (but this is fattening, so limit it).
For some of the above reasons, many people do not keep them in community tanks or planted tanks (some people keep them in bare tanks just to ease cleaning). I decided to try it anyway in my 75gal tank, since I heard mixed things about keeping them with other fish. The water changes were done pretty frequently anyway, and the heat did not bother the other fish in that tank (I kept the temp around 82 F). I did not have problems with most of the discus, though one did suffer from HITH and had to be treated separately for a while.
Eventually, some of them started pairing off and starting to mate, which I wasn't ready for in the community tank. Because it was so difficult to keep them in breeding condition, I decided to sell off the young adults. Old Blue (who was always alone; he had lived for years with his previous owner that way for some time) is today my last discus, and I'm sad he doesn't have companions, but I think he otherwise behaves as a discus should. He is a very beautiful fish, and probably the largest fish I have now out of all my tanks, something of the crown jewel in my 75gal tank.
Discus breeding, I've heard, is somewhat like angel breeding, but having never tried it intentionally, I will refer you to the other sites below for details. One interesting factoid about them is that they actually feed their young with a milky slime they produce on the sides of their body!