Several actors made their name in show-business by being the understudy of a leading actor and taking the role over for several performances (an example being, Sir Anthony Hopkins with Sir Laurence Olivier, when Olivier became ill with cancer during the run of the National Theatre's The Dance of Death, 1967).
More recently, the term "understudy" has generally only been applied to performers who will cover a part, but still regularly perform in another role within the show; usually a minor, extra role. Performers who are only committed to covering a part and do not regularly appear in the show are often referred to as standbys. Standbys are normally required to sign-in and remain at the theater the same as other cast members, although sometimes they may call in, until they are released by the Production Stage Manager. If there is no doubt about the health of the actor being "covered," or there are no hazardous stunts to be performed, a standby may be released at the first intermission if not before. At times, standbys are required to stay within a certain area around the theatre (10 blocks in New York City is a common standard). The standby must also have a cell phone so that at any time they can be called to the theatre.
The term is also regularly applied in association football, most often to describe a second choice goalkeeper whose appearances are limited to when the regular first choice player is absent through injury or suspension. A notable example is Chris Woods as long term understudy to Peter Shilton for the England national football team.
In musical theatre, the term swing is often used to refer to a member of the company who understudies several chorus and/or dancing roles.
In the opera world, the term used is cover or covering.