The notion forgetful functor is not precisely defined and there are various discussions on what should be considered a forgetful functor (forgetting structure vs. forgetting properties). When used the term is used colloquially and should simply point out that the functor is the result of the human ability to forget things. It immediately describes what the functor does (e.g., the forgetful functor from $Ab$ to $Grp$ forgets that the group you have is abelian (forgetting a property) while the forgetful functor from $Ab$ to $Set$ forgets that you have a group and just remembers the set of elements (forgetting structure)).
Now, since what matters in a category are the morphisms and not the objects, saying 'forgetful functor' may be a bit subtle. For instance, there is a forgetful functor from the category of frames to the category of complete semi lattices even though the objects are identical. What is being forgotten here is the fact that morphisms of frames respect arbitrary joins as well as finite meets while morphisms of complete semi lattices preserve arbitrary joins (and may or may not preserve finite meets). The two categories, while having the same objects, are vastly different.
Another example is the category of frames and of locales. This time there is no forgetful functor even though, again, the categories have the same objects. The category of locales is the opposite of the category of frames, so it is very difficult to forget things when you are inverting the direction of arrows.
To conclude, 'forgetful' is not precisely defined. A forgetful functor is very often assumed faithful, but you can concoct a situation where you forget stuff and somehow loose faithfulness. Since forgetting is in the eye of the beholder, or the mind of the observer, (nearly) everything is possible.