Why does magnetic force only act on moving charges? I don't understand why the magnetic force only acts on moving charges. When I have a permanent magnet and place another magnet inside its field, they clearly act as forces onto one another with them both being stationary. Also, I am clearly misunderstanding something.
 A: As mentioned in a comment to the question: in terms of relativistic physics there is a way to address your question.
In terms of pre-relativistic physics it is something that must be granted in order to formulate a theory of electromagnetism. Phrased differently: in terms of the physics prior to the introduction of relavistic physics the fact that magnetism is proportional to velocity has to be accepted as is.

There is a set of notes by Daniel V. Schroeder titled:
Magnetism, radiation, and relativity
Preliminary remark:
The electric field is extremely strong. It is far stronger than we would tend to expect, since in daily life we never experience that. The effects of the kind of static electric charge we can feel in daily life are minute because there is never more than the tiniest difference of charge.

In these set of notes by Daniel Schroeder the following is pointed out: a current flowing through a wire is a population of electrons that is moving with respect to the wire.
In terms of pre-relativistic theory of electricity and magnetism: the wire is neutral everywhere because the density of positive charges and negative charges is the same everywhere.
Now in terms of relativistic physics:
We take the wire as a whole as stationary reference. The population of free-to-move-around electrons in the wire then has (on average) a velocity with respect to that stationary reference. For that velocity there is a corresponding length contraction. So in that sense the wire and the electron population do not have the same charge density.
Schroeder shows that it is possible to account for magnetic effects with this difference in charge density.
That is counter-intuitive on several levels, of course. For one thing: it is always asserted that relativistic effects require relativistic velocities in order to be measurable at all.
The (average) velocity of the electron population is very, very small. The thing is: the Coulomb force is so mindbogglingly strong that magnetism can indeed be accounted for in this way.

(One thing that concerns me though is that in those Schroeder notes relativity of simultaneity is not discussed. I suspect the Schroeder notes present an oversimplification. The numbers check out, so Schroeder is doing something right, but we can't be sure of what it is that is done right.)
A: A magnetic field does have an influence on a stationary electric charge.

"when i have a permanent magnet and place another magnet inside its field they clearly act a forces onto one other with them both being stationary"

is exactly the case that is also true for an electron.
An electron is both an electric charge and a magnetic dipole. Any description of internal currents in the electron to create the magnetic dipole is superfluous. It is quite sufficient to consider the magnetic field of the electron to be just as fundamental as its electric field.
For a detailed description of why an electron moving through a magnetic field is deflected laterally (Lorentz force, Hall effect(s) ), see here.
A: After reading some of the other answers I think there is a simpler more fundamental explanation.
According to Field Theory, rather than considering the interactions between particles and field, we can consider the interactions between the fields associated to the particles.
Under this understanding, Classical gravitation $F = G \frac{m_1 m_2}{r^2}$ is the interaction between 2 objects, each with it's own gravitational field.  Similar Coulomb force $F = k\frac{q_1 q_2}{r^2}$ is the interaction between 2 charged particles, each with it's own electrical field.
Magnetic interactions are a bit more complicated, but we can also understand magnetism as interaction between objects that each have their own magnetic field.
Now to your question.
A static electric charge generates an electric field, but not a magnetic field, as per the relevant Maxwell equation. $\vec\nabla\cdot \vec E = \frac{\rho}{\epsilon_0}$.
A moving charge has a magnetic field, given by $\vec\nabla\times \vec B = \mu_0 ( \vec J + \epsilon_0 \frac{\partial \vec E}{\partial t})$. in which the current density $J = q\cdot \vec v$
Therefore a magnetic field does interact with a moving charge, because the moving charge also has a magnetic field, but not with a static charge, because the static charge does not have a magnetic field.
A: This is basically just a matter of definition. The magnetic force is not something independent or complete on its own. Instead, it is the electromagnetic force that is the underlying concept. The electromagnetic force can be split into a portion that acts on stationary charges and a portion that acts on moving charges. We call the first portion the electric force and the second portion we call the magnetic force. But they are not independent. They exist together and it is just an essentially arbitrary division to split the overall electromagnetic force into an electric force and a magnetic force.
Regarding a permanent magnet: a permanent magnet is typically uncharged, so you would not consider a permanent magnet as though it were an electric charge. It would typically be better to consider a permanent magnet as an uncharged magnetic dipole. Inhomogeneous magnetic fields can exert forces on stationary magnetic dipoles. The formulas are not at all the same for magnetic dipoles as for electric charges.
A: As a simple(ish) special case of the principle set out by @Cleonis, imagine two initially stationary charges, starting a short distance apart.  Under the Coulomb (electrostatic) force, they will accelerate apart (if they're like charges) or together (if they're opposite charges).
Now think about this same pair of charges from the point of view of an observer who is moving perpendicular to the displacement between the two charges, so that, to this observer, the charges appear to be initially moving parallel to each other and perpendicular to the spacing between them.
Because the displacement between the two charges is perpendicular to the frame-of-reference motion, it will not appear Fitzgerald-contracted.  However, there will be time dilation, so that to this observer, it will appear to take longer for the two charges to reach any given distance apart than it did to the stationary observer; i.e. the acceleration of the charges will appear to be smaller to the moving observer than to the stationary observer; i.e. (again) the force between the charges will appear to be smaller to the moving observer than to the stationary observer.  That is, an observer who encounters moving (in the same direction, perpendicular to the spacing between them) charges will think there is an extra force between them, acting in the opposite direction to the Coulomb force, compared with an observer who encounters stationary charges.  That extra force is what we call the magnetic force.
A: At an effective classical level, the atoms in permanent magnets do contain moving electric charges at the microscopic level: the orbiting electrons. These moving charges correspond to microscopic electric currents, and the magnetic fields act on these microscopic currents.
This picture is certainly a simplification of the underlying quantum effects, but I think it's accurate enough to clear up your confusion.
A: To be more precise, the magnetic force acts on currents. A moving particle implies a current, although not all currents correspond to a classical, visible movement. This is the case with the microscopic currents in permanent magnets.
