Title says it all. I've been looking at K-groups of a few C* algebras. My very rough understanding is that these groups reflect (in some appropriate sense) the algebraic structure that we get when we take direct sums of projective modules. This much I have understood by calculating the K0 groups of the complex numbers, complex polynomials in one variable, and cuntz algebras. In the first two cases, the only projective modules are free, so the semi group we get is the natural numbers. I know for cuntz algebras, weirder things happen when you take direct sums of projectives.
I would appreciate if someone could illuminate exactly what the K0 group of a C*-algebra tells us about the structure of that algebra.