A vector space is a field $~$F$~$ paired with a Group $~$V$~$ and a function $~$\cdot : F \times V \to V$~$ (called "scalar multiplication") such that $~$1 \cdot v = v$~$ and such that scalar multiplication distributes over addition. Elements of the field are called "scalars," elements of the group are called "vectors."

Also there are some nice geometric interpretations and stuff.