Geometric meaning of the determinant of a matrix

If you think about the matrix as representing a linear transformation, then the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed. So for instance, the matrix $2I$ stretches a square of area 1 into a square with area 4, since the determinant is 4. This idea works in all dimensions too, not just 2 or 3!

This also translates well when you get in to more general mappings $f:\Bbb{R}^n\rightarrow\Bbb{R}^n$; if the function is nice enough, you can represent it "locally" by a linear transformation. The (absolute value of the) determinant of this linear transformation gives the "local" volume distortion of the function - i.e. how much the function is stretching or compressing regions of space near a point.