Where do magnetic monopoles get their magnetic fields from? As I understand it, magnetism isn't a "real" force like electricity is – when you apply relativity to a moving charge, it produces an effect that can be described using magnetic fields, but it is, in fact, just the electric field behaving relativistically. If this is the case, then what is a magnetic monopole? I'm assuming it's not secretly an electric charge moving relativistically, but if that's how we define a magnetic field, then what does it actually mean for a magnetic monopole to have a magnetic field?
To be clear, I understand that Maxwell's equations would be modified to accomodate the monopoles without a current. But in terms of relativity, what is fundamentally happening?
 A: Your intuition is almost right, but not quite. Yes, when the electric field of a charge is described in a moving frame, magnetic fields appear. But that does not make the magnetic field less fundamental.
Why a charged object produces an electric field? Well because it has electrons and protons that by themselves have also an elementary electric charge (no more questions asked here). This elementary charge is a fundamental quantity.
Where does a magnet get its magnetic field from? Partially from moving charges, but some of its magnetism also comes from stationary electrons and protons (and even neutrons) as well. These subatomic particles have an intrinsic magnetic moment (basically they behave as a femto-sized magnet). This elementary magnetic moment does not come from anything more fundamental as far as we know. It is analogous to the elementary charge.
You can be in the rest frame of a neutron (that has no net charge) and interact with the magnetic field of its magnetic moment. In a moving frame, electric fields would appear. In this case electric fields are the relative effect, not the magnetic field.
You also have to consider electromagnetic waves, where you can have an oscillating electric and an oscillating magnetic field in vacuum. Which is more fundamental? None, you cannot move to a frame where one or the other disappears as EM waves travel at the speed of light.
What about magnetic monopoles? If the monopoles exist, we will not be able to understand them as anything more fundamental than the elementary charge, or the magnetic moment of the electron. If there are elementary particles with magnetic monopoles, then this would be just another elementary property. Just because it is related to magnetic field does not make it more unusual than the electric charge.
Summary: magnetic fields are not more real than the electric fields. And a magnetic monopole would be just be as fundamental as an elementary electric charge.
A: I would like to add a few things to what Mauricio said.

Why a charged object produces an electric field?

A macroscopic object emerges as a charged object after a charge separation. One partial volume has an excess of electrons and the other an excess of protons. The volumes are also completely separable into two objects.

Where does a magnet get its magnetic field from?

A permanent magnet obtains its magnetic field through the collective alignment and permanent self-retention (below Curie temperatures) of the electrons, protons and neutrons involved. In an electromagnet - with the help of an electric current - the magnetic dipole polarisation of the involved subatomic particles takes place.
A magnetic field cannot be divided into a northern and a southern object by dividing the object into two volumes. What you get are again dipoles. If you make this division down to the atoms, you get the magnetic dipoles inherent in the subatomic particles.

What about magnetic monopoles?


*

*in the direction of absolute zero, magnetism and electricity do not change. Towards high energies, electricity and magnetism may merge into a uniform mush.

*IF magnetic monopoles existed, how would they differ from electric charges? We could separate them like electric charges, they would flow as a current. There is no difference and therefore no indication of whether it is an electric charge or a (hypothetical) magnetic charge.

*BUT, we will not be able to separate them because of a nuance. Besides the chemical separation of charges, only electromagnetic induction separates electric charges. And only charges accelerated in loops induce magnetic alignments. By what do you separate magnetic monopoles?

So for me, the intrinsic properties of electric charge and magnetic dipole in subatomic particles are fundamental and the more interesting question is how much more fundamental particles can be used to model electric field, magnetic field and EM radiation.
