TITLE: In what sense is the forgetful functor $Ab \to Grp$ forgetful?
QUESTION [5 upvotes]: One sometimes hears about "the forgetful functor $Ab \to Grp$." Given that the image of an object under this functor is still abelian, in what sense is this "forgetful"?

REPLY [2 votes]: Given that "forgetful functor" does not have an agreed upon definition, this question is somewhat soft, and perhaps should be tagged as such.
So all this is a matter of individual taste.
As we can read in Wikipedia generally, but not always,  "forgetful" functors are faithful and have left adjoints.
However, the functor $Ab \to Grp$ should be called "the inclusion functor", because it is not only faithful - like most forgetful functors - but also injective on objects. And this is a distinguished characteristic of "inclusion functors". You can read this here.
Because of this, we say that $Ab$ is a subcategory of $Grp$. 
Since this particular functor is also full, we say that $Ab$ is a full subcategory of $Grp$.
Also the other functors mentioned by @Ittay: $Fld \to Ring$, $ComRing \to Ring$, $ComMon \to Mon$ are inclusion functors.