Among the standard examples of abelian categories without enough projectives, there are

1. the categories of sheaves of abelian groups on a topological space (as VA said), or sheaves of modules over a ringed space, or quasi-coherent sheaves on a non-affine scheme;
2. the categories of comodules over a coalgebra or coring.

No abelian category where the functors of infinite product are not exact can have enough projectives.  In Grothendieck categories (i.e. abelian categories with exact functors of small filtered colimits and a set of generators) there are always enough injectives, but may be not enough projectives.

Among the standard examples of abelian categories with enough projectives, there are

1. the category of functors from a small category to an abelian category with enough projectives (as VA said), or the category of additive functors from any small additive category to the category of abelian groups (this class of examples includes the categories of modules over any rings);
3. the category of pseudo-compact modules over a pseudo-compact ring (see Gabriel's dissertation);
4. the category of contramodules over a coalgebra or coring (see Eilenberg-Moore, "Foundations of relative homological algebra").