"Proper" is an adjective used to describe a morphism of spaces—topological spaces, schemes, locales, etc—that is sufficiently nice and has some neat properties. Between topological spaces a morphism is proper if the preimage of compact set is compact, which is a clean definition, but it doesn't give me any geometric intuition that generalizes to schemes or beyond. Similarly the definitions I've seen for a proper morphisms of schemes hasn't helped improve my intuition.
What are some flags/heuristics/features of a morphism that I should look out for to decide whether or not it's proper? What are some illustrative examples or non-examples of proper morphisms, in whatever setting, that could help build my geometric intuition here? This answer over on MathOverflow helps a bit, but only a bit,