An matrix can be seen as describing a linear map in dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of -dimensional space.
For example, a matrix , seen as a linear map, will turn a square in 2-dimensional space into a parallelogram. That parallellogram's area will be times as big as the square's area.
In the same way, a matrix , seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be times as big as the cube's volume.
The determinant can be negative or zero. A linear map can stretch and scale a volume, but it can also reflect it over an axis. Whenever this happens, the sign of the determinant changes from positive to negative, or from negative to positive. A negative determinant means that the volume was mirrored over an odd number of axes.
One can think of a matrix as describing a system of linear equations. That system has a unique non-trivial solution exactly when the determinant is not 0 (non-trivial meaning that the solution is not just all zeros).
If the determinant is zero, then there is either no unique non-trivial solution, or there are infinitely many.
A matrix has an inverse matrix exactly when the determinant is not 0. For this reason, a matrix with a non-zero determinant is called invertible. If the determinant is 0, then the matrix is called non-invertible or singular.
Geometrically, one can think of a singular matrix as "flattening" the parallelepiped into a parallelogram, or a parallelogram into a line. Then the volume or area is 0, which means that there is no linear map that will bring the old shape back.