For questions involving the forgetful functor, the functor that drops some of the properties of the input structure before mapping to the output.

Forgetful functors are a generalization of the canonical morphisms over algebraic structures. Given a group/ring/field/topological space $A$ and normal subgroup/ideal/sub-field/compact subset $B$, we can define the forgetful map $$A\to A\:/\:B$$by mapping every element to its representative in the quotient. In some sense, this map "ignores" some of the structure of $A$ in this map, reducing $A$ to a "simpler" structure.

Forgetful Functors generalize this notion to categories, where we replace the idea of a quotient by an essentially surjective (a functor such that every element in the output category is isomorphic to an element outputted by the functor) and full (the induced homologies are surjective). In some sense, these conditions ensure that the input category is somehow "richer" than the output category, and that some of its structure is preserved by the functor.

Use this tag if your question involves a forgetful functor.