Before there were theaters, there were touring acting companies. These companies did not have a building in which to perform their plays, so they toured their regions and played wherever they could rent space. Most of the time, that happened to be in the courtyards of inns. The companies would erect their stage at one end of the courtyard and the inn's residents would either stand around the stage or go out on their rooms' balconies and watch from there.
The Theater was the first permanant theater in England, and was built by James Burbage. The shape and form of his theater was taken directly from the inns in which the acting companies performed. The Theater was eventually torn down and The Globe Theater was created from its parts. The same form was used, which is why it was shaped in a sort of semi-circle, with tiers of balconies all around and space in front of the stage in which people could stand to watch the plays.
Modern stages take a few different forms. The one that might closely resemble The Globe Theater would be the thrust stage. This stage juts out into the audience, so the audience might be sitting on the three sides surrounding it. Another type of stage is the arena stage, in which the stage is in the middle and the audience sits on all sides of it. But the most common is the proscenium stage. This is what most high school auditoriums have. There is no outer stage, as in the thrust and arena types, and a curtain separates it from the audience. It's like looking into a picture or a window.
Hope that helps!