A forgetful functor is a functor which is defined by ‘forgetting’ some structure. For example, the forgetful functor from Grp to Set forgets the group structure of a group, remembering only the underlying set.
In common parlance, the term ‘forgetful functor’ has no precise definition, being simply used whenever a functor is obviously defined by forgetting something. Many forgetful functors of this sort have left or right adjoints (and many are actually monadic or comonadic), leading to the paradigmatic adjunction “free forgetful.”
On the other hand, from the perspective of stuff, structure, property, every functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties). From this perspective, the forgetful functor from to forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the free group functor) is also forgetful: if you identify with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.
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