A free module is rather intuitive: there exists a subset satisfying suitable properties which may act as a basis for elements. Taking as example an $\mathbb R$-module, or a vector space, we see "free" as a generalisation of this concept.
However, a module $P$ is said to be projective if for every surjective homomorphism $\epsilon : B \to C$ and homomorphism $\gamma : P \to C$, there exists a homomorphism $\beta: P \to B$ with $\epsilon\beta = \gamma$.
When I say am looking for "intuition" regarding this definition, I mean I am looking for this: since we can say a free module is a generalisation of the notion of a vector space with a basis, then a projective module can be seen as a generalisation of what?
If such a comparison can't be made, is there some alternate way of understanding the definition, perhaps more concretely than a notion in homological algebra?
I am aware of the Serre-Swan theorem, however, I am not sure this sheds much light, since a projective module is not really a generalisation of a vector bundle on a compact manifold. They are separate things, related by the theorem.