Learn French: Une ombre bouge derrière la porte. - A shadow is moving behind the door.

You would replace une ombre (a shadow) with personne (nobody) and place ne immediately before the verb. The sentence becomes: Personne ne bouge derrière la porte. Notice that when personne acts as the subject, it goes at the very beginning of the sentence.

Does the verb bouger apply to any kind of "moving"?

Bouger specifically means making a physical movement or stirring, just like the shadow does here. If you want to say you are moving to a new house or apartment, you must use déménager. If you are talking about traveling from one place to another, you would use se déplacer.

I always mix up the French words for "behind", "in front of", "before", and "after". Which one is derrière?

Derrière is used strictly for physical space, meaning "behind". Its exact opposite is devant (in front of). If you are talking about time, you use avant (before) and après (after). Because the door is a physical object here, we use derrière.

Does une ombre just mean "a shadow"?