It's used in several contexts.
In the saturday morning cartoons, it was a common voice over and editorial comment by Stan Lee, and also in internal monologues by Peter Parker. In Amazing Fantasy #15, it's also a narrative context.
It's attributed in later canon to Uncle Ben, both shown as Ben speaking it to Peter in the car, and also Peter in narrator mode (especially in the cartoons) mentioning it in internal monologue, usually in the form of, "As Uncle Ben said, 'With great power comes great responsibility.'"
It has also been used on occasion by other characters; I don't remember exactly where, but ISTR Professor X has used it as an occasional admonition to students at the Xavier School.