In film and video, production refers to the part of the process in which footage is recorded. This is what most people imagine when they think of a film being made — actors on sets, cameras rolling, etc. The production phase is also known as principal photography.
In large feature films the beginning of the production phase marks the "point of no return", i.e. the point at which it is no longer financially viable to cancel the project. At this point it is almost always cheaper to continue until the project is finished than to deal with the financial fall-out of canceling.
The goal of principal photography is obviously to record all required shots, however it is fairly common to shoot "pick-up" shots in post-production. Pick-up shots may be required when a mistake is noticed, a script change is made (this is unusual), or even if a performance is deemed to be unsatisfactory.
In music, production usually refers to the creative direction of a project. Unlike a film producer who is more of a manager, a music producer has a very hands-on role in the creative development.