Name for diagonals of a matrix

$\diagdown$ Major, Principal, Primary, Main; diagonal $\diagdown$

$\diagup$ Minor, Counter, Secondary, Anti-; diagonal $\diagup$

The diagonal from the top left corner to the bottom right corner of a square matrix is called the main diagonal or leading diagonal.

The other diagonal from the top right to the bottom left corner is called antidiagonal or counterdiagonal.

You can also define the main diagonal and antidiagonal of a rectangular matrix. See here for more.

The general term for any diagonal going top-left to bottom-right direction is $k$-diagonal where $k$ is an offset form the main diagonal.

$k=1$ is the superdiagonal, $k=0$ is the main diagonal, and $k=-1$ is the subdiagonal.

According to Mathworld, the general term for the antidiagonals seems to be skew-diagonals.