In Aristotle's Poetics, the text explains his theory of tragedy. (As a side note, Aristotle's defining of tragedy and the tragic hero is still used today.)
According to the text, plot is the most important aspect of the tragedy. Within this, Aristotle defines plot as "the arrangement of the incidents." This refers to Aristotle's idea that the utmost importance lies in how the scenes are played out for the audience.
The plot, therefore, must be complete (including everything it needs to insure both arrangement and unity). This is where "unity of action" comes into play. Any tragic play must rely upon its own movement, each scene relying upon the previous and setting up the next. The play, essentially, must be self reliant. The play cannot rely upon any outside forces to insure its unity of action. These outside forces are called deus ex machina (where some sudden change (new character or ability) solves the problems of the play).
An example of this is seen in William Shakespeare's tragic play Macbeth. The unity of action relies upon the intertwining of all of the actions by the characters; Macbeth is the central figure which insures unity. Essentially, the witches' prophecy leads to Macbeth's rising ambition. This rising ambition (his hamartia, or tragic flaw) leads to Macbeth's murder of Duncan. Duncan's murder leads to his son's fleeing the country. When Duncan's sons flee, they plan revenge upon Macbeth. This plan forces Macbeth to insure his crown is kept (through more murder).