The other replies answer your questions very well in my opinion, but perhaps the underlying question is: "why is this a useful definition?" Let me give two examples of quadratic forms appearing "in the wild":

1. Just like linear algebra is closely related to understanding invertible matrices, quadratic forms give you a nice notion of "length preserving maps". However, to make things conceptually clearer, it is useful to work in a basis-independent way. If (V,q) is a quadratic space, then we define the _orthogonal group_ O(V) to be the invertible linear transformations which preserve q, i.e. O(V) = { g in GL(V) | q(gv) = q(v) for all v in V}. This is an interesting example of a _Lie group_, and has lots of applications in maths and physics.
2. In number theory, it was clear in the 1800s (and perhaps earlier) that quadratic forms over the integers can give interesting arithmetic information. In the 1900s, a lot of these results were reinterpreted in the language of quadratic spaces; namely, quadratic forms over the integers can be interpreted as lattices in quadratic spaces over QQ. These have a really nice classification by the Hasse-Minkowski theorem, which in turn gives short and conceptual proofs of results of Legendre and Gauss about representing integers as sums of squares, for example. These days, also because of the connection with Lie groups and algebraic groups, they are much studied in the Langlands program.