In stage script speak, a beat is the term used to indicate that the actor should pause while reading a line. This tends to carry more weight than the pause that would be connotated from a comma or a period - while the other two are merely punctuation marks, used to indicate the flow of the line, a beat emphasizes the empty space. When a writer wants the character to express a change in their train of thought or emotion, or if they want to leave space for reactions, they will make use of a beat. A beat lasts for different lengths, depending on the writer, but the general idea is that a beat lasts around the same length as a long breath would.
Other words which, in a script, mean something similar (but not quite the same) as a beat include:
-Pause, which usually signifies a longer and heavier break in the dialogue than a beat.
-Silence, which usually is much longer than a pause.
-Space, which is much rarer and really can't be generalized.
Of course, writers being as finicky as they are, there are no absolutes in these definitions. Some writers use "pause" to mean just a simple, short pause, and others don't use any wording other than normal punctuation.
Perhaps more commonly (even though the term "beat" as described above is nearly universal in stage plays), a beat can be used to refer to an important moment inside of a script. In both writing for the stage and writing for the screen, the writer begins with a list of the beats of their script - the "beat sheet" - and then begins writing their script out from there. In American movies, beats tend to fall at a rate of roughly one beat every five minutes.