How does Shakespeare's theatre compare to modern theatre?

Theatres in the Elizabethan Era (Shakespeare's time) were very different to modern theatres for a number of reasons. Firstly, the buildings were very different: the roof was often thatched, there was an open yard for poorer people to stand and watch the performance and the stage was much wider. The stage at Shakespeare's theatre, the Globe, for example, measured 44 feet in width. 

Secondly, the actors were very different. Elizabethan women were legally barred from working in the theatre so female characters were played by male actors. There were also more parts than there were actors so speeches had to be included in the script to indicate that an actor was now playing a different role. 

Finally, the plays themselves were very different. Plays were often written by anonymous writers or by members of the clergy and not by well-known people, as is common today. In addition, in the early Elizabethan period, many plays were Greek or Roman stories and were performed in the original language so the spectators would not have understood the dialogue word-for-word. 

I think you are asking about the physical theaters in which Shakespearean plays were first performed, and how these might compare to the theaters in which we'd expect to see the plays now. Shakespeare's plays are most associated with the Globe Theatre in London, which was actually the first purpose-built playhouse of its kind and was financed by Shakespeare's acting troupe.

Prior to the building of The Globe, plays were regularly performed in any space that was large enough to host them, be it a village square, a pub, a schoolhouse, or a royal court. The Globe, then, has far more in common with our modern theaters than it did with the "theaters" of the time before it. Still, it differed somewhat from the modern understanding of a theater.

In the first place, it had no roof, being open to the air and shaped like a Roman amphitheatre. Only the stage had a roof ("the heavens") for the purposes of hanging scenery from it. People would sit all around the circular theater on the raised banks of seats, but there was also a "pit" in front of the stage where people could stand, for a smaller fee than a ticket with a seat. There was also a balcony above the stage, probably for musicians.