There are many grades and types of cotton fabric aroung. Simply speaking higher the thread weight count, the tighter the weave of the shirt. You will often see shirts listed as an 18 single, or 30 single, or 40 single. The higher the number, the finer the weave of the shirt, and the more lightweight it will be. Generally, nicer shirts are 30 single. 40 single would be more like a vintage type shirt- very lightweight. An 18 single would be more heavy weight, or a coarser weave.

Pre-washed means that it helps the shirt not to shrink further. You definitely want pre-washed or pre-shrunk, especially if you are purchasing 100% cotton.

Combed cotton is just a type of very soft cotton. WIthout going into the long definition of it, it just means that before the cotton is spun into thread, it is combed and any short or impure fibers are removed. This makes it softer and more costly. Combed cotton would be preferable.

A lot depends on the type of t-shirt you are looking to make. If you are trying to make a sports shirt, softness may not be that much of a factor. If you are trying to make a fashion shirt, softness would be a much larger factor.