What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $n \times n$ matrix by breaking it up into the determinants of smaller matrices. Apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related?

(I considered making this a comment, but I thought it might deserve more attention than a comment would receive. Upvotes and downvotes will tell if I am right or wrong).
Complement about the sign of the determinant
I loved the accepted answer by Jamie, but I was frustrated that it did not give more explanation about the sign of the determinant and the notion of "rotation" or "orientation" of a vector. The answer from Marc Van Leeuwen comments more on this, but maybe not be enough for everyone -- at least not for me -- to understand what it means for a matrix to change the orientation of the space it transforms. So I googled the issue and ended up on the following explanation which I find excellent and accessible:
http://mathinsight.org/determinant_linear_transformation#lintrans3D
