What does it mean for a matrix to be orthogonally diagonalizable?

I assume that by $A$ being orthogonally diagonalizable, you mean that there's an orthogonal matrix U and a diagnonal matrix $D$ such that

$$A = UDU^{-1} = UDU^T \text{.}$$

A must then be symmetric, since (note that since $D$ is diagnonal, $D^T = D$!) $$A^T = \left(UDU^T\right)^T = \left(DU^T\right)^TU^T = UD^TU^T = UDU^T = A \text{.}$$

A square matrix is said to be orthogonally diagonalizable if there exist an orhtogonal matrix $P$ such that $P^{-1}AP$ is a diagonal matrix.

A square matrix $A$ is orthogonally diagonalizable $\Leftrightarrow$ $A$ is symmetric.