The base layer (usually man made fibre) draws sweat away from the body (wicking) and allows the moisture to evaporate rather than turning into a puddle of sweat. In colder weather more layers are added, the second insulating layer is usually something fleecy, with the outer water/wind proof layer atop that.
Don't run in cotton T shirt, you will finish your run soaked and your t shirt twice as heavy!
I have various base layer type T's: Helly Hansen and Sugoi probably my favs. You can always buy relatively cheap non branded gear which is just as good I'm sure