Discoveries
In the final scene of The Tempest, there is a climactic and dramatic moment. Prospero speaks to Alonzo, King of Naples, who believes that his son, Ferdinand, was drowned in a wreck at sea, and promises "a wonder to content ye" (5.1.171). The stage direction then reads:
Here Prospero discovers Ferdinand and Miranda playing at chess.
Clearly, the two actors playing the parts of Ferdinand and Miranda were already playing the game, presumably sitting at a table, since we then hear the first words of their conversation before they realize that others are looking at them. How are they "discovered"?
Early reconstructions
In early reconstructions of the Globe theater, it was assumed that there was something called an "inner stage," separated by a curtain from the rest of the stage, where intimate action took place; discoveries would be behind the curtain. But there is much evidence that argues against an inner stage of this kind (most notably the fact that the highest-paying customers would see none of the action), and it is more likely that the scholars who proposed this solution were simply trying to make the Elizabethan theater more like the Victorian stage, with its curtained proscenium arch.
The solution proposed more recently, and followed in the design of the New Globe, is that there was a central doorway, perhaps one of double width, which could be curtained, and which could serve both as an additional entrance for major characters, and a discovery space.
There was certainly a curtain of some kind for characters to hide behind -- often called the "arras": in Hamlet, Polonius and Claudius spy on Hamlet from behind the arras as he is with Ophelia, and later Polonius is stabbed through the arras by Hamlet.
An alternative possibility is that one of the two normal doors was used for discoveries, and that there was simply a curtain hanging in the space between the two doors, with room enough for an actor to stand behind it without causing it to bulge suspiciously. In any case, it is certain that the plays were sometimes staged with only two doors, when the company went on tour, or performed in the home of a nobleman, since they would have had to play on old-fashioned stages there.