When I was a student, I once watched a professor (a famous and brilliant mathematician) spend a whole class period proving that the functor $M\otimes-$ is right exact. (This was in the context of modules over a commutative ring.) He was working from the generators-and-relations definition of the tensor product. With what I'd consider the "right" definition of $M\otimes-$, as the left adjoint of a Hom functor, the proof becomes trivial: Left adjoints preserve colimits, in particular coequalizers. Since the functors in question are additive, $M\otimes-$ also preserves 0 and therefore preserves cokernels. And that's what right-exactness means.