TITLE: How to interpret Hessian of a function
QUESTION [7 upvotes]: I know that gradient of a function gives the direction in which the directional derivative of the function is maximum. Is there any similar interpretation of Hessian ?

REPLY [8 votes]: This is more a comment to Brulboy's answer than an answer in itself. 
If you fix a point $x$, you can usually${}^{[1]}$ diagonalize the Hessian matrix with an orthonormal basis. This means that you can take a Cartesian orthogonal coordinate system centered at $x$ which makes your matrix look like this: 
$$Hf(x)\equiv \begin{bmatrix} \lambda_1 & & &\\ & \lambda_2 & &\\ &  & \ddots &\\& & &\lambda_n\end{bmatrix}.$$
Those $\lambda$s are second derivatives of functions of a single variable. Specifically, if you restrict $f$ to the $j$-th coordinate axis (of the diagonalizing coordinate system), you get a function of one variable whose second derivative is exactly $\lambda_j$. 
Therefore, when you look at a Hessian matrix at a point and see a mess of numbers, you are simply not choosing the best coordinates for that point.

${}^{[1]}$. By usually I mean: even if somebody proved that, sometimes, this cannot be done, the examples are somewhat pathological and unlikely to manifest themselves in real life. The matter is discussed thoroughly here.