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?

In addition to the answers, above, the determinant is a function from the set of square matrices into the real numbers that preserves the operation of multiplication:
\begin{equation}\det(AB) = \det(A)\det(B) \end{equation}
and so it carries $some$ information about square matrices into the much more familiar set of real numbers.
Some examples: 
The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)
Which real number does not have a multiplicative inverse?  The number 0.  So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.
What is the determinant of the inverse of a matrix?  The inverse of the determinant, of course.  (Etc.)
This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.
