This list is going to be absolutely massive. There are literally thousands of fish that people put in aquariums, so I will list some of the most common by family and a few characteristics.
This ageless aquarium favourite has now been bred to include black, white, red and patched varieties as well as the timeless rusty gold. Goldfish are coldwater fish, and can tolerate a wide range of pH, temperature and salinity, making them ideal for beginners and low-tech setups.
My personal favourites, the family Poecilidae includes guppies, Endler's livebearers, swordtails, platies and mollies (as well as less colourful wild species.) The female livebearer gives birth to up to a hundred fry every month, which are freeswimming from birth. They are easy to breed and I'm currently overrun with platy fry.
Guppies especially are very hardy and tolerant, and are favourites for children and beginners. Platies are also very tough and are prettier than guppies in my opinion. They are my favourite fish. Swordtails, mollies and endlers are touchier and require more complex setups.
This catfish family is a big favourite for keeping a large aquarium clean. They are active schooling catfish, very easy to breed. The albino variety in particular is very attractive. Pygmy varieties have been bred
The plecostemus is HOT. That's the only way to put it. Most of these fish grow very big - around 30-40cm - but there are plenty of plecos suitable for large community aquariua. Plecos are catfish. The bristlenose catfish is a small pleco species.
Angels can be delicate, and are not good for beginners. They can also be aggressive, and will eat tetras. They are very attractive, and a good addition to a community aquarium that does not contain tetras if kept alone.
These small, colourful fish are very attractive and very much favourites in a community aquarium. They are not very tough, and it's easy to kill them with a sudden shock like a temperature change or a pH jump.
For a complete list far better than I can provide:
http://www.auspet.com/fish08.HTML - I so owe oxy for that link!!