Unfortunately, that looks like a common goldfish. These grow to 12 inches long, and do best in ponds or tanks in the 50-100 gallon range. Stunting is a serious problem with goldfish that are housed in too small tanks, so the sooner you can upgrade, the better.
Goldfish generally need at least ten times filtration, as they are huge waste producers and very oxygen hungry. You're going to need to really watch your water parameters, as the amount of ammonia one goldfish can produce in an uncycled tank is astounding.
If you don't have much in the way of funds, a large sterilite tub with a good filter should suffice for now. You can chuck in some fast-growing plants like duckweed or elodea to suck up excess nitrates/ammonia, although your goldfish might appreciate the snack instead.
The most important thing now is to get him into a much bigger tank/tub, and to start getting it cycled. This will at least keep him healthy and comfortable, and take some of the stress off you.