In theatre (especially in the illusionistic Western tradition), breaking character occurs when an actor ceases to maintain the illusion that they are identical with the character they are portraying. If the breaking of character is particularly serious, it is considered corpsing, which in film or television would normally result in an abandonment of that take.
Examples
Catherine Schell, who found it difficult to act with Peter Sellers in The Return of the Pink Panther and maintain her composure; several scenes showing her laughing at his antics remain in the film.
Many instances of breaking character have occurred on Saturday Night Live, such as a sketch where Christina Applegate and David Spade could not stop laughing at Chris Farley's motivational speaker character, Matt Foley, as well as the band members in the "More Cowbell" sketch reacting to Will Ferrell's antics. Jimmy Fallon often broke character, which became one of his trademarks.