What is that which is magnetic force?

One model of a magnet (typically, ferromagnet) is the Ising model, which basically places an electron with spin at every atom in the structure of the magnet. Although the magnet is not charged overall, nor is it moving, there can be motion of the atoms (and electrons) that make up the magnet. In this model, the spin of the electron produces small magnetic moments which are aligned, and sum to a large magnetic moment.

So permanent magnetic materials are termed ferromagnetic. It is a good question to ask what the nature of the force is between two magnets, but I believe the answer is: we don't know.

Take as a similar type of example: electric charges. We know that like electrical charges repel one another and unlike charges attract, and we have an equation which we can use to calculate the resultant force (Coulomb's law).

But as to why it is so that like charges repel and unlike charges attract, we can only say that the fields created by these charges interact in such a way as to make this true.

The same with magnets. The magnetic field surrounding magnets will interact in such a way as to make like poles repel and unlike poles attract.

Perhaps someone who is versed in QED can tell us if that theory shines any stronger light on this.

I'm mentally visualizing a water balloon between two magnets. What is the water balloon composed of? Is it purely the "magnetic field"? If so, seems like more work needs to be done.

The interaction between electric (magnetic) fields from two sources is described by virtual photons. Virtual means, not existing in reality. So the real interaction between such fields is explained only in a statistical way and not in detail.

As long as the inner structure of these fields isn’t subject of interest, as long a detailed description is not possible. Why this is of an interest? Because, if one would think about an inner structure of such fields, there is no way not to think about some components or quanta. These quanta have to have some extent and energy content. Ergo electric and magnetic fields have to be finite. This would be in contradiction to the current perceptions.

The magnetic force can be understood as a relativistic effect on the electric force. It results from Lorentz transformation of the electrostatic force for a stationary charged particle.