The ellipsis indicates a lacuna, an omission; ellipses show that something has been left out.

The idea is to not change the meaning of a quotation, for instance, by omitting text and inserting an ellipsis to mark that lacuna.

A number of style guides (and check with your publisher and editor) allow ellipses to be used to indicate a pause in dialog, or a speaker trailing off in mid-thought. Chicago makes a distinction between an ellipsis used for "faltering or fragmented speech" and a dash used for more substantial pauses (for instance).

Don't . . . be . . . . like . . . Barbara Cartland. You have a limited non-renewable lifetime allocation of ! and . . . ; don't use them rashly.