The major types of stage used in modern theatre are:
Proscenium stage: This is a rectangular space framed by an arch or "proscenium: (literally, something in front of the "skene"). It is almost as if you took and ordinary room and removed one of the four walls. The spectators sit outside the imaginary "fourth wall" and look in. Much of realistic drama uses proscenium stages.
Theatre in the Round: This is a variant on traditional Greek staging, in which the stage is circular and the audience sits around the circular performance space. Many directors prefer a variant of this known as "three-quarter round" (basically just like ancient Greek theatres) as it makes for easier staging.
Thrust Stages: Essentially a proscenium stage with a semi-circular platform extending out into the audience.
Found Spaces: Some theatre companies use experimental or found spaces (such as in street theatre or plays produced in churches).