An animated character can talk, cry, yell, scream or whatever, but while his mouth may undergo any number of contortions, the point of his chin never moves, and the general shape of his face never changes.
It takes a lot of time, effort and artistic skill to animate characters whose faces stretch and warp realistically when their mouths open and change shape. Therefore, as a cost- and labor-saving measure, most of today's animated characters in all but the highest-end productions (or, of late, in CGI animation) are constructed with static heads/faces and a series of mouths that can be overlaid on them. This wasn't always true - the older cartoons had full facial warping with speech, for example - but with the general decline in American animation starting in the 1950s, such shortcuts became standard practice. The exception in a cartoon which otherwise does this exclusively is profile shots; these are (usually) animated with full-face movement because otherwise it looks much stranger than the animators want.
For similar reasons, the number of shapes required to animate a character's Mouth Flaps is kept to a minimum - some studios get away with as few as three to six mouth shapes (closed straight line/smile, closed mouth, small open, medium open, large "O", and an optional "screaming" mouth).
An alternative to animating the mouth at all emerged in the 1960s with the development of "Synchro-Vox", a technique whereby the mouths of live actors were superimposed on the faces of drawn characters. (The cartoon Clutch Cargo remains the foremost example of this dubious development. Well, that and Conan O'Brien.) In addition to reducing the animators' efforts (and thus production costs) to nearly nil, Syncro-Vox had the added benefit of turning what might have been an ordinary cartoon into a surreal experience that could inspire nightmares in an entire generation of children. This was parodied in the extra film on The Incredibles' DVD.
See also No Knees.
Since this trope is so ubiquitous, please limit examples to exceptions in 2-D animation.