You have a few choices with a tank that small. Take a look at this thread:
http://www.aquariumadvice.com/viewtopic.php?t=69218&highlight=stock++gal
You could get a single male betta, which are gorgeous fish with great personalities. With a 6-gallon you could also maybe get a couple of otos for the bottom too as a cleanup crew. That would be the limit probably.
Or you could get a small school (around 4 or 5) of little tetras, like neons or glowlights. They would be very pretty in the tank.
Note in the linked thread above, that with tropical fish you will need to get a little (25-watt) heater and thermometer to make the temp stable at 78 to 80 degrees. You will also need a mini gravel vacuum, bucket, and dechlorinator for water changes, if they didn't already sell you these things. And you will need a test kit.
You might want to post a separate question about the best way to start up a 6-gallon. Small tanks are harder to cycle and keep stable.
Cycling is the process by which you make the water safer for your fish--it involves growing colonies of bacteria so the ammonia in the fish urine won't kill them. Not cycling is why new fish tanks most often fail and people give up on fishkeeping in frustration. There are good articles explaining cycling on this site. Very small tanks are difficult to cycle--when I kept my 2-gallon tank, I just did large water changes very frequently to keep ammonia down. But cycling is THE most important thing you can do to keep your fish safe in a larger tank. I think I have seen people here who are successful cycling 6-gallons. You should talk to them and see what they say.
Cycling is hard on fish, and tetras and some of the other options for small tanks (see above linked topic) are pretty fragile. People here who keep small tanks might have some good tips on how to start your tank safely, how often and how much water to change at first, and what regular maintenance afterward requires.
Good luck, and keep asking questions!