Why does a magnetic field reorient another magnet along its direction? I mean, what is the force that causes it? Is it the Lorentz force? 
 A: I mean, is there any other EM force? Funny thing, if you model the magnets as point dipoles, their interaction is conservative, form a well defined potential, and this potential is minimised when they are aligned in the same direction (assuming their magnetic moment direction is the same as the direction of their separation):
So yes, the interaction is caused the Lorentz force. It is caused since one dipole $\mathbf m$ creates a magnetic field in point $\mathbf r$ (as taken from here), note that electric and magnetic dipoles are completely analogous,
$$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi r^3}\left[\ (3\mathbf m\cdot\hat{\mathbf{r}})\hat{\mathbf r} - \mathbf m\ \right]$$
Interaction between a dipole and a magnetic field is described by,
$$U=-\mathbf m\cdot\mathbf B\ \text{ and }\ \boldsymbol\tau = \mathbf m\times\mathbf B$$
where $U$ is the magnetostatic potential energy and $\boldsymbol\tau$ is the torque acting on it. As usual,
$$\mathbf F=-\nabla U$$
gives you the total force that this dipole feels in this system. Since $U(\mathbf r)$ is not constant in $\mathbf r$, its gradient will not be $0$, the dipoles will exert force to each other. Since the potential is electromagnetic in nature, so is the resultant force (and thus it must be the Lorentz force).
The thing is, you can also write the force equation using some more convoluted math, but the argument above amounts to the same.
A: According to Ampère all magnetism is caused by electric currents.
This is obvious for an electromagnet,
where the current through a coil of wire
produces the magnetic field.
It is also true for a bar magnet.
You can imagine the bar magnet as a collection of
many microscopically small current loops
(the electrons orbiting around their nucleus
and electrons spinning around themselves).
So in both kinds of magnets the magnetic field
is produced by current loops.
The force between currents (i.e. moving charges)
is essentially the Lorentz force.
Parallel currents attract each other,
and antiparallel currents repel each other.
This mechanism is enough to explain the attraction and repulsion between magnets.

(image from Encyclopedia Britannica: Repulsion or attraction between two magnetic dipoles)
It also explains why a magnet feels a torque
when tilted by an angle in the field of another magnet.

