A vector space is a place where you have things called vectors, and what you can do with them is to add them and stretch them. You also need to tell when some of them are equal. In order to tell who got added to whom, vectors are usually drawn as lines with an arrow at the tip, according to which the rule is: start at a tail, go to the head and if there is a vector to be added, put its tail there and continue. Of course, a lot of things are implicit in such an informal definition, such as that vectors can be moved around without essentially changing them, that there ought to be a single place called an origin where vectors get started off, and if they are just laying around and happen to cross, that doesn't count.
Addition is commutative, in the sense that you reach the same destination whichever order is followed, something called a parallelogram law. It is also assumed that stretching is related to addition, in the sense that 
and so on for all the constructions related to integers, rational numbers, real factors and even going so far as complex numbers.
It is one thing to work up an axiomatic theory of something, like groups, or topologies, or vector spaces, or whatever. It is quite another to discover that all of those systems and indeed almost every theory that one can imagine, have very similar structural features, for which there is an applicable metatheory, sometimes called Universal Algebra. That higher level theory is based on the ideas of equivalence and order relations, classifying the functions mapping one set to another, and the generation of new structures via the intermediary of cartesian products. To appreciate this view of vector spaces, consider: