A projective module is a module over a ring that satisfies the property that for any surjective module homomorphism, there exists a module homomorphism that makes the following diagram commute: 
In simpler terms, a projective module is one that has the ability to "split" a surjective homomorphism, meaning that it can be mapped onto by another module homomorphism.
A projective module is different from a free module in that it does not necessarily have a basis, whereas a free module always has a basis. This means that a projective module has more structure and constraints than a free module.
This result is significant because it allows us to understand the structure of projective modules better. It also helps us to classify and compare different types of modules, as well as prove other important theorems in abstract algebra.
The proof involves constructing a basis for the projective module and showing that it satisfies the necessary conditions for a direct sum. This includes proving that the basis elements are linearly independent and span the module, as well as showing that the direct sum of the submodules generated by the basis elements is isomorphic to the original module.
Yes, the result can be extended to other types of modules, such as injective modules and flat modules. This is because the proof relies on the universal property of direct sums, which is a general concept in abstract algebra. However, the specific approach and techniques used may vary depending on the type of module being considered.