What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$? I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. 
To me this implies that the electron doesn't orbit in a disk like fashion, it precesses. Is this correct? Is there any further significance?
Also, what's the total angular momentum $J$? how is this related to $\ell$? Why doesn't $\ell$ give the total angular momentum?  
 A: The value $m_{l}$ is the eigenvalue of the operator $L_{z}$ determined by seeing the action of this operator on the eigenstate $|~l,m_{l}>$ or in other words $$ L_{z} |~l,m_{l}> = m_{l} |~l,m_{l}> $$
While $l$ is related to the total angular momentum operator $L$ and it acts on the same eigenstate giving you
$$ L^2 |~l,m_{l}> = l(l+1) |~l,m_{l}> $$
The relations between the two operators is given by
$$L^2 = L_x^2 + L_y^2 + L_z^2 ~~~~\mbox{and}~~~~ \vec{L} = (L_x, L_y, L_z)$$
The eigenvalues are also related and the relation that you can find in almost any textbook about quantum mechanics is 
$$ - l \leq m_l \leq l ~~~~\mbox{all of them interspaced by unity}$$
The idea that the electron is moving around the atomic nucleus is a simplification.
The electron is not localized, all the information you can have about his position is expressed in a probabilistic form: the probability of finding the electron in the position $(r,\theta,\phi)$ is given by $|<r,\theta,\phi~|~\psi>|^2$ assuming that the wavefunction $|~\psi>$ is normalized.
In the case of hydrogen atom we have $|~\psi> \propto|~l,m_{l}> $ or in the coordinate rappresentation  $$(<r,\theta,\phi~|~\psi>= \psi(r,\theta,\phi) ) \propto (<r,\theta,\phi~|~l,m_{l}>= Y_l^m(\theta,\phi)) $$
what you can see in the left hand side of this equation is nothing but the square root of
the probability I stated above and this tells you that the probability depends on the spherical harmonics $Y_l^m(\theta,\phi) $ which also depends on the $m_l$ value.
So you can imagine that a the $m_l$ tells you the shape of the spherical harmonics which gives you the probability of finding the electron at some point in space. That's the best of the orbits you can get. 
The operator $J$ is the total angular momentum which is expressed as $J = L + S$ and that's just a definition that comes from the fact that particles have also spin angular momentum. The relation of the eigenvalues of $J$ (which are just $j$) with $l$ and $s$ has a quite long derivation that you can find for instance on the book Quantum Mechanics by Auletta Fortunato Parisi .
Anyway the relation is $$ |l - s| \leq j \leq |l+s| $$ all of them interspaced by unity.
