Two Latin square of the same size are said to be orthogonal if you form a square by superimposing the two squares in the following way,
$\left[\begin{array}{l}1&2&3\\3&1&2\\2&3&1\end{array}\right]*$
$\left[\begin{array}{l}1&3&2\\3&2&1\\2&1&3\end{array}\right]=$
$\left[\begin{array}{l}1,1&2,3&3,2\\3,3&1,2&2,1\\2,2&3,1&1,3\end{array}\right]$ you get all possible pairs as entries , i.e all entries are distinct. My initial guess was that this somehow captures the idea of forming new latin sqaures by permuting rows and columns of a given latin squares.So that two squares are orthogonal if they cannot be converted into each other by permuting rows or columns. But this is not true as you can see,
$\left[\begin{array}{l}1&2&3\\3&1&2\\2&3&1\end{array}\right]*\left[\begin{array}{l}1&2&3\\2&3&1\\3&1&2\end{array}\right]=\left[\begin{array}{l}1,1&2,2&3,3\\3,2&1,3&2,1\\2,3&3,1&1,2\end{array}\right]$
I am not able to find the motivation behind defining the orthogonality condition on Latin squares.