Psuedo-Inverses seem to be rarely used by pure mathematicians, but to be of great use. They allow us to solve matrix equations where true inverses don't exist. In other words, they allow us to generate a pre-image of a vector under a mapping, even if that mapping wasn't one to one. The seminal work, at least to the extent that I can discover, seems to have been done (seperately) by Kalman and Penrose.
While the details can be somewhat grueling, the basic idea is quite elegant and simple. Suppose we have a non-square
matrix A, say of dimension m by n. Now, let us suppose that either the columns or rows (whichever has the lower
dimension, of course) are independent. Then consider: either 
(m by m) or 
(n by n) will be a square
matrix with full rank, and thus invertible. In the latter case, we say that
is the psuedo-inverse, since
Similarly if m < n, though we will need to post- rather than pre-multiply. For this reason, these two are often called, respectively, the Left and Right psuedo inverses, corresponding to what's known in Linear Algebra circles as the Over- and Under-Determined Cases. It can be shown that these psuedo-inverses always exist, and, with some additional conditions, are unique. Also, if the true inverse exists, the psuedo-inverse simplifies to it.