There are far too many common uses of vector spaces to list. It's hard for me to remember what it's like to think without using vector spaces. For anybody who'd like an introduction, I suggest the Wikipedia article http://en.wikipedia.org/wiki/Vector_space.
Probably the single most comprehensive use of vector spaces is in quantum mechanics. In the QM of any isolated system, all physical states are simply vectors in a particular space. The time derivative of the state vector is given by a linear operator on the vector.
To take a particularly simple example, the single-particle Schrödinger equation represents the single-particle states as wave-functions, ψ(r
,t). We can call the function at any time, t, a vector |ψ> in a space consisting of a set of such functions. The time-dependent Schrödinger equation just expresses the time dependence of that vector as a constant times a linear operator (iH), where H is called the Hamiltonian, on the vector. d|ψ>/dt=-iH|ψ>/h_bar. For certain special state vectors, H|ψ>=E|ψ>, where E is just a number. In vector language, these are called eigenvectors of the linear operator H. These are the famous quantum states of definite energy.
Since the hope is that gravity will some day be included with the other fundamental interactions in the quantum mechanical formalism, it is possible that someday we will say that all of fundamental physics is represented by a vector space.
(published on 12/12/10)