In dialogue, to show faltering, fragmented speech, interrupted speech, or instead of a dash to grab the reader's attention.
In narrative, to show omission of quoted material.
There are different ways to indicate an ellipsis:
Three periods with spaces: Robbi once used . . . oh, how I remember . . . an ellipsis correctly.
Three periods with spaces before and after: Robbi never uses ... and I mean never ... ellipses correctly.
Some references also allow for three periods followed by a space: Robbi... and I know this to be true... rolls her eyes every time I correct her incorrect use of ellipses.