What do I need to make a basic gym in my apartment?

If you want to keep it simple maybe consider some free weights, exercise mat, and a jump rope. Throw in some decent exercise DVDs and you can get quite a bit done using your own body and just a few basic pieces of equipment. Some people might add an exercise ball, resistance bands, etc.. You can find a vast number of fairly inexpensive items to add to your list.
You could go all the way to the other end and purchase an all-in-one gym style equipment designed for home use. A treadmill or an elliptical are other things at the higher end of the spectrum.