Also, it is known that isomorphism of vector bundles translates to Murray-von Neumann equivalence of the associated projection in K ⊗ C(X), where K is the compact operators on a separable Hilbert space.
The key reason for the introduction of K-theoretic methods into the study of C*-algebras was the Fredholm index: Given a bounded linear operator on a Hilbert space that has finite-dimensional kernel and cokernel, one can associate to it an integer, which, as it turns out, reflects the 'defect' on the operator - i.e. the extent to which it is not invertible.
The Fredholm index map appears in the 6-term exact sequence given by the Calkin algebra.
Later on, Brown, Douglas and Fillmore observed that the Fredholm index was the missing ingredient in classifying essentially normal operators up to certain natural equivalence.
This, in turn, led to K-homology, Kasparov's bivariant KK-theory, and, more recently, Connes and Higson's E-theory.