>>10025418Grothendieck defined Grothendieck groups, not K-theory.
K-theory is a cohomology theory, which means (amongst many other things) we have a group K^i (X) for each i.
K^0 (X) is always the Grothendieck group of something (vector bundles, coherent sheaves, projective modules, etc.), but the higher K-groups may be different depending on the type of K-theory we are talking about.
Complex Topological K-theory is probably the "easiest" form of K-theory, in the sense the higher K-groups are always either K^0 (X) = (Grothndieck group of complex vector bundles on X) or K^0 (?X) where ?X is the suspension of X.
But for instance if you do Algebraic K-theory, the higher K-groups are much more complicated.
