A square matrix M=m_ij is called symmetric if M^t = M (it is equal to its transpose), or, equivalently, m_ij = m_ji, or, equivalently, that for all vectors v,w, (Mv,w) = (v,Mw). This last property shows that a symmetric matrix stays symmetric in any orthogonal basis! (Note: Previously this last sentence was incorrectly phrased!)

n*n real symmetric matrices (symmetric matrices of real numbers) are important because they are guaranteed to be diagonalizable, i.e. they have n eigenvectors forming a base (or, equivalently, they have n eigenvalues if you count eigenvalues with their (geometric) multiplicity).