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Many stage plays - including all thirty-seven/thirty-nine of Shakespeare's - consist of five acts, while others consist of only three acts. (Are there any other common numbers of acts?)
What's the difference between a three-act play and a five-act play, in terms of the kind of story they tell and how they tell it? What considerations tend to be taken into account by the playwright when deciding how many acts to divide their play into?
This is an excellent site discussing five-act structure in Shakespeare, containing a diagram and numerous examples. Essentially, the five-act structure can be broken down as:
Exposition of the five-act structure is associated with Freytag, who was concerned with Classical Drama and Shakespeare. The link to criticism of Freytag discusses how structure changes with modern literature. The compaction of late acts is probably most brilliantly demonstrated by Hitchcock in the ending of North by Northwest.
The idea of the three-act structure can be said to originate with Aristotle, who believed plays needed to have a beginning, middle and end.
Indiana University has an excellent page discussing it here which breaks down the sections as:
With each section divided by a plot point.
It is important to note that the three-act structure is mostly associated with story arcs in cinema in the contemporary landscape. Wikipedia lists it exclusively as a structure utilized in film and television. Despite the potentially hyperbolic title of this page, it is nevertheless an extremely insightful and useful analysis.
The reason you won't find much on three-act structure in plays is because number of acts in any given play is quite variable. Modern and contemporary playwrights, dating to at least Ibsen, experiment with different act structures. A good playwright can find a way to structure their story regardless of the act structure, as is demonstrated by many works considered masterpieces with structures beginning at one act and proceeding upwards from there.
Contemporary dramaturges regularly alter the "breakup" of longer plays, as the contemporary theater tend to avoid productions with more than a single intermission. Thus a "five-act" Shakespeare play is commonly presented in two halves, with a single intermission.
I put "five-acts" in quotes because this structure is a later innovation, not present in Elizabethan, or even Jacobean, drama in any formal sense.
Regarding alternate structures:
One-act is a very common form, and arguably the original form, deriving from Greek Drama. Aristotle considered this a requirement for greatness in drama. [See the Classical unities]
A famous examples of a modern one-act is Wilde's Salome, but there are many such examples, and it's a quite popular contemporary structure because of it's efficiency from the standpoint of time. Becket heavily utilized a one act structure as he reduced drama to it's most basic form. Breath has a run-time of about 25 seconds.
Modern act structure was in large part driven by theater concessions. Thus Maxwell Anderson or Noel Coward might write a four act play—four acts mens three intermission during which the audience may visit the bar. This slowly transformed into two act plays as people became busier, (and possibly attention spans shortened as new media such as film and television became dominant,) and the acceptable length for the average play got shorter.
However, Angels in America is an example of an extraordinarily successful play unconcerned with running time. From a link to a current production of the play: "Run Time: Millennium Approaches runs approximately three hours. Perestroika runs approximately three hours and 40 minutes."
It is notable that in cinema, where the three-act structure dominant, this structure is applied to work that does not feature an intermission, or any temporal division, between the acts, except in rare cases where a film may have an intermission. (The "Roadshow" edit of Hateful 8 which has a running time of 187 minutes, is a recent example of a film presented with an intermission.)
