Probably its most common uses in dramatic films are to adjust the pace of the main action, to conceal the deletion of some unwanted part of the main shot, or to allow the joining of parts of two versions of that shot. For example, a scene may be improved by cutting a few frames out of an actor's pause; a brief view of some listener can help conceal the break. Or the actor may fumble some of his lines in a group shot; rather than discarding a good version of the shot, the director may just have the actor repeat the lines "in one" and cut to that solitary view when necessary--some actors have fumbled their lines deliberately to get that treatment.
These are journeyman techniques. Cutaways can also be used for reasons of art.
One example of a cutaway being used deliberately to break continuity, for comedic effect, appears in Shoot the Piano Player (Tirez sur le pianiste), by François Truffaut: the pianist and his female companion are being followed; she opens her compact and uses it to show him the two gangsters behind them, impossibly large in the reflection.
Cross cutting is a series of cutaways and cutbacks.
In news and documentary work, the cutaway is used much as it would be in fiction. On location, there is usually just one camera to film an interview, and it's usually trained on the interviewee. Often there is only one microphone. After the interview, the interviewer will usually repeat his questions while he himself is being filmed, with pauses as he pretends to listen to the answers. (The comedy Micki + Maude shows this process quite accurately.)These shots can be used as cutaways. They may be necessary just to ensure that the audience can hear the questions correctly.