What's the difference between a vector space and a field? They seem to have the same definitions. The only difference is that a vector space is over a field (what exactly does this mean?)
A vector space is an "abelian group" over a field, meaning the "scalars" are elements of a field.
If you never studied Field Theory, there is no way you can possibly understand what I just said. The reason why I am saying this is perhaps you are learning Linear Algebra and they mention "field" briefly but you do not understand. If that is the case I cannot explain any better. The meaning of a field happens to be a very very long explanation.