A singular matrix is one whose rows are linearly dependent. A Gauss-Jordan elimination on a singular matrix will leave you with one or more lines full of zeroes.
Another way to look at it: suppose your matrix A is a representation of some linear system of equations (each row would correspond to one equation). If you have less equations than unknowns, you can solve by choosing arbitrary values for these "orphan" unknowns. Meaning you have infinite solutions (or no solutions at all).
There's many different ways to prove that a matrix is singular given those conditions. One of them could be assuming that A in NOT singular and then testing for linear independence. Another way could be using determinants: you could factor out a 0 and cancel every cofactor in the expansion.
If the general form nxm gives you trouble, try working it out for a 2x2 and then expand on that.