In earlier days, the film editor would assemble the movie by splicing individual pieces of film together. When run through a camera, the spliced film then formed a complete sequence.
These days, film editing is often down through computer software, which allows the editor to move much more quickly and keep track of individual shots more readily.
Films often go through numerous "cuts" or edits. The first cut is called a rough cut and it usually longer than the final film so that the director or studio can make changes more readily.
Once a rough cut is complete, the director can collaborate with the editor on a more refined cut. When that process is complete, the studio gives it a final okay (or asks for more changes), resulting in a final cut of the film.
Soviet film theorists postulated that an inferred meaning exists between individual pieces of film cut together, which contributes to the overall effect. By shifting the order of the shots, you change its overall meaning, a process know as montage.
Continuity refers to the natural progression of shots and the internal logic in their assembly that lets the audience know what's going on. It's sacrosanct for most Hollywood films though experimental films sometimes deliberately disrupt continuity in order to achieve a particular effect.