One example in which diagonalization is important is the study of quadratic forms.
http://en.wikipedia.org/wiki/Quadratic_form
A quadratic form can be written as
Q(x) = x^TAx
where x is a column vector and A is a symmetric matrix. It is a theorem that you can always diagonalize A by a rotation of coordinates. For example, in 2D, if you have an equation such as: ax^2+2bxy+cy^2 = D, then by rotating your coordinate axes you can rewrite the equation as
A\bar{x}^2+B\bar{y}^2 = D
in your new coordinates. Therefore, the original equation represents an ellipse or a hyperbola (or possibly a pair of parallel lines if one of the eigenvalues A or B is zero.)
Quadratic forms are important, for example, because a general function f(x,y,z) has a local Taylor polynomial approximation
f = f(P) + df + Q_f + higher order terms
The second order term is a quadratic form which is determined by the Hessian matrix. So, for example, at a critical point (where the differential df =0), the first nonzero term in \Delta f is the quadratic form determined by the Hessian. Since all quadratic forms can be diagonalized by a rotation of coordinates, that means that by a rotation of coordinates,
\Delta f = A\bar{x}^2+B\bar{y}^2+C\bar{z}^2 + higher order terms
A, B, C are the eigenvalues of the Hessian. One thing you can do with this knowledge is determine whether a critical point is a maximum. To do that you check the eigenvalues of the Hessian matrix. If they are all negative, then you have a relative maximum.
