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An object P is projective if Hom(P, −) preserves epimorphisms, equivalently, if every morphism P → C/B lifts to a morphism P → C. Projectives are dual to injectives and generalize free modules. In Module categories, projectives are direct summands of free modules. Every object has a projective cover, a surjection from a projective object with 'minimal kernel'.
From your study of free objects, you know that free modules have a remarkable property: given any surjection M → N and any map F → N from a free module F, there exists a map F → M that makes the diagram commute — the map can always be "lifted." Projective objects are defined by exactly this lifting property, generalized to arbitrary additive or abelian categories without requiring the object to be free in any literal sense.
The definition is this: an object P is projective if for every epimorphism e: C ↠ B and every morphism f: P → B, there exists a morphism f̃: P → C such that e ∘ f̃ = f. The map f̃ is the lift — it reaches through the surjection and lands in the "larger" object C rather than the quotient B. Equivalently, the functor Hom(P, −) preserves epimorphisms: whenever C → B is surjective, the induced map Hom(P, C) → Hom(P, B) is also surjective. This is the categorical dual of the definition of injective objects, where it is Hom(−, I) that preserves monomorphisms. Projective and injective objects are mirror images — the duality of "surjections lift in" versus "injections extend out."
In the category of R-modules, projective modules are precisely the direct summands of free modules: M is projective if and only if there exists a module N such that M ⊕ N is free. Over a field, every module is free and hence projective. Over ℤ (a principal ideal domain), every projective module is in fact free — the two notions coincide for PIDs. Over more general rings, projective-but-not-free modules exist and are geometrically meaningful. The finitely generated projective modules over the ring of continuous functions on a compact space are exactly the vector bundles over that space (Serre-Swan theorem). A vector bundle that becomes trivial when you add a trivial bundle is algebraically a projective module that becomes free when summed with a free module — the projective condition captures stable triviality.
The projective cover of an object M is a surjection P ↠ M from a projective object P such that the kernel is superfluous — removing it cannot produce a smaller projective surjecting onto M. Projective covers are the minimal projective objects that map onto M, and they provide canonical minimal projective resolutions: exact sequences 0 ← M ← P₀ ← P₁ ← P₂ ← ··· where each Pᵢ is projective and as small as possible. These resolutions are the raw material for derived functors such as Tor and Ext — which measure how far a functor deviates from exactness and encode deep information about the module structure of a ring. Not every abelian category has projective covers (sheaves often lack them), but in the module categories where they exist, minimal projective resolutions are the canonical tool for homological computation.