abelian category so that we can speak of short exact sequences in (the category of chains in ). We can then define a covariant endofunctor to be "exact" if a short exact sequence is carried to a short exact sequence . Or, equivalently we are saying that the induced functor carres SES to SES.
Now, note that is an abelian category. One can then prove that is projective if and only if the (covariant) Hom functor is exact where (as I'm sure you know) acts on the objects of by sending and acts on morphisms by sending to where .
You can find more information on this in any good algebra book, and in any good homological algebra book. Perhaps the most self-contained, elementary discussion of this can be found here in Rotman.