The vector nature of Angular Momentum
Here is an animation from Wikipedia : http://en.wikipedia.org/wiki/File:Torque_animation.gif
The angular momentum is given by the Cross product of r and p We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation) I don't think this really makes sense, how is the vector nature of angular momentum justified ? How can one get an intuitive sense about the direction of angular momentum ? 
Re: The vector nature of Angular Momentum
I think that direction of angular momentum is only a convention. Someone thought of that rule, it seemed practical since intensity of cross product of r and p really defines its amplitude. I am not sure, but think you could define another rule and get same physical results.
Such thing is with phasor diagrams, you choose phase of one quantity to be zero, and according to that do everything else. This is only my opinion. 
Re: The vector nature of Angular Momentum
Having angular momentum as a vector comes in handy when you want to explain gyroscopic precession.
Maybe someone with more knowledge than me can give better examples :) 
Re: The vector nature of Angular Momentum
The vector comes into play when you're trying to solve for conservation of angular momentum. The vectors must all add to zero.

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If you want to think about it some more, you might ask, why the right hand rule, why not a left hand rule? You could use either, and everything would still make sense. That's because the angular momentum vector is not truly a vector, it is a "pseudovector", one that depends on which hand you use. Well, the laws of physics don't depend on which hand you use, and true vectors don't depend on which hand you use, so true vectors are, in a sense, more "real" than pseudovectors. For calculation purposes, pseudovectors are nice, just three components that transform almost like a vector. But when you want to do theoretical work, you might not want to deal with the artificiality of pseudovectors. The bottom line is that pseudovectors are better represented by antisymmetric 3x3 matrices (antisymmetric tensors). Instead of a pseudovector [x,y,z] you use [tex]\left[\begin{matrix} 0 & z & y \\z & 0 & x \\ y & x & 0 \end{matrix}\right][/tex] This tensor transforms the same way no matter what, no worry about which hand you need to use, and its better for theoretical work, its the "real thing", unlike the more concise pseudotensor. 
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I guess a good thing to think about is how else you would do it? 
Re: The vector nature of Angular Momentum
angular momentum and torque while having vectorial nature are pseudovectors i.e. under an inversion they don't get reflected.
L=r×p replacing r by r ,gives inversion.In p=m dr/dt,it becomes .so overall L does not change direction. edit:oops,someone else also written it. 
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