A linear transformation T from an n dimensional space to itself (or an n by n matrix) is singular when its determinant vanishes. This means that there is a linear combination of its columns (not all of whose coefficients are 0) which sums to the 0 vector. (Since the n dimensional parallelopiped formed by them has no volume.)
This means that the same linear combination of the basis vectors is mapped by the transformation into the zero vector. A vector that is mapped into the 0 vector by a transformation is said to be in the kernel of that transformation. Since the transformation is linear any linear combination of vectors in its kernel are in it as well, and the kernel is a subspace of the domain of the transformation. Vectors that are normal to every vector in the range of T form the null space of T. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity of the transformation. A singular transformation is one with a non-zero nullity.
The same considerations apply to rows as well as columns. If M is singular there must be a linear combination of rows of M that sums to the zero row vector. That same linear combination of the column basis vectors must be perpendicular to every vector in the range of M, so that its transpose must be in the null space of M.

Proof

Thus if M is singular, all of its range is perpendicular to at least some vector and the dimension of the range cannot exceed n-1. The dimension of the range of M or T is called the rank of the transformation or matrix.
You can find the rank of a matrix by row reducing it; the number of non-trivial rows that do not vanish as you row reduce is the rank of the matrix. The number of rows of zeroes that you are stuck with at the end is the nullity of the matrix.
The rank plus the nullity of an n by n matrix is n.

Example

M has rank less than n or  non zero nullity are both synonyms for M being singular.