What's a quadratic map and how do I invert it?
In cryptographic applications a quadratic map is generally conceived as a system of n quadratic equations in n variables.  Each equation is generally represented as an n-long vector of n variables x and a matrix Mi (for equation i of the system.  The equation is then:

(x^T)Mi(x) = ai

Your are then given the n matrices Mi, and a resultant vector of the ai.  The inversion you are looking for is to find a vector x such that:

(x^T)Mi(x) = ai for all i

If the Mi are chosen at random then the problem of finding a solution to this problem is believed to be intractably hard (NP-hard in fact).  If the Mi are specially chosen (not random at all) one can make inversion easy.  Mulivariate Quadratic cryptography tries to take an easy to solve quadratic map and transform it into a seemingly difficult to solve map.

If you look at the Unbalanced Oil and Vinegar schemes:

https://en.wikipedia.org/wiki/Unbalanced_Oil_and_Vinegar

you get an idea of one such system.