A field is an algebraic category in which the elements are an abelian (commutative) group under addition and F/{0} is also an abelian group (this time) under multiplication. The most common field in most people's experience is the set of rational numbers. What this generally boils down to is that every non-zero element has a multiplicative inverse (i.e. division by zero is not allowed.)

A vector space has two operative sets, the vectors and the scalars (the field) where there is a "multiplication" defined between the vectors and the scalars) by a*v = av, a "new" vector. This is called scalar multiplication. There is also an "addition" operation between vectors called vector addition which also yields a "new vector". The scalars (the field) maintains all the addition and multiplication properties of a field.

The easiest example is probably ordered pairs (x,y) (the vectors) and the real numbers, (the field of scalars) which incorporate all the operations in entities called linear combinations, such as 4*(1,-3) + 3*(5,1) = (4,-12) + (15,3) = (19, -9). I could go on forever, but I think you can go on from here and talk about dimension (here it is 2) and bases (here, for example, {(1 , 0), (0,1)}.