TITLE: Why is it useful to classify the vector bundles of a space?
QUESTION [32 upvotes]: It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces.  Now, I've read somewhere that it is an important and difficult problem to classify all of the vector bundles of a space.  I'm willing to accept that the problem is difficult, but why is it important? What are some applications of such a classification?

REPLY [4 votes]: I suppose a simpler answer to your question is that the classification of vector bundles is to vector bundles as Whitney's embedding theorem is to manifolds.  
Specifically, Whitney's weak embedding theorem says that all n-manifolds embed in $\mathbb R^{2n+1}$ and that any two embeddings of an $n$-manifold in $\mathbb R^{2n+2}$ are isotopic. So studying the "abstract" problem of $n$-manifolds up to diffeomorphism is equivalent to the "less abstract" problem of studying $n$-dimensional submanifolds of $\mathbb R^{2n+2}$ up to isotopy. 
Whitney's proof of the above fact is almost exactly how the proof of the classification of vector bundles works.  Moreover, they're philosophically almost identical proofs, as the Whitney theorem says abstract manifolds are submanifolds. The classification theorem says abstract vector bundles can be thought of as having their fibres in Euclidean space.