Explain the origin of magnetic fields in layman terms Electromagnetism always seemed so weird to me. It feels like it should be 2 different things entirely. At first I thought magnetism was just applied electric charge, and that a magnetic dipole was just an electric dipole on a net neutral macroscopic object. However I learned this was false when I learned a magnetic field can be created by an electron, which has electric monoploe.
Electric fields I at least understand the origins of. When there are 2 charges in space, they apply a force on one another. The net effect of those forces on any one charge at any given point in space can be represented mathematically with the electric field. 
But what force of attraction/repulsion, in that case, is a magnetic field representing? It's apparently still caused by the same electromagnetic force, but it follows entirely different rules, so different that a partical with an electric monopole can have a magnetic dipole.
I understand a magnetic field is created when an electric charge moves, as well as by the particle spin of a charged particle. But is that all we've got on its origins? Does it just show up when electric charges move and that's that? Why is it considered the same fundamental force as electric attraction/repulsion when it follows wildly different rules? CAN it be derived from the rules of electric charges?
Please understand, when you explain it, that you're talking to an undergraduate chemical engineer here, not a physics major. Please try to explain it while keeping physics terminology and math beyond high school level to an absolute minimum.
Edit: Once again, I want to stress that I know next to nothing about wave functions, the concept of symmetry, and angular momentum is something I've only kind of learned about and only kind of understand at an intuitive level. (My basic understanding of angular momentum is that it is a constant change in linear momentum as governed by a force perpendicular to the velocity of the object at any given time. I understand this could be technically wrong).
Not saying I'm incapable of understanding these concepts if you explain them to me in layman's terms. I'm not dumb. I just know next to none of the vocab.
 A: 
At first I thought magnetism was just applied electric charge.

I don't know what you mean by this statement, if you could expand it in another post, it would be helpful. Applied in what way? Electromagnets is all I can guess at here, sorry.

But what force of attraction/repulsion, in that case, is a magnetic field representing.

There is a way of combining electric forces and magnetic forces, in the  Faraday tensor $F_{\mu\upsilon} $  Keeping this in layman terms, it means what one person says is an electric effect, is viewed by another person, moving at some velocity relative to the first guy, as a magnetic effect. I haven't the space to expand on this, but an Internet search should give you details at various levels of detail.
So electricity and magnetism are pretty directly connected, even if we can have electric monopoles  but we have never found magnetic monopoles. Another example of the interconnection are the Maxwell Equations which you could search for more details .

I understand a magnetic field is created when an electric charge moves, as well as by the particle spin of a charged particle. But is that all we've got on its origins? Does it just show up when electric charges move and that's that? 

As far as ultimate origins go, there are a few things we have to accept. The electric charge on an electron is a physical constant we can measure, but AFAIK,  there is no fundamental theory to expand it. I am deliberating avoiding the electroweak force, as not being layman friendly.
The magnetic charge we can link to something, as you know, the spin of the electron. (which is an totally abstract idea, again this is not news to you, an electron is not a tiny revolving soccer ball like thing).  We have an equation that links the tiny magnetic force (moment) of  the electron to its  spin, but why the spin exists is not understood, it's only treatable  as a nature given physical constant.
When electrons move, they do generate electromagnetic fields and we can predict the strength and  direction of these fields, and we have a mathematical method that (sort of, imo) explains why they are connected , and definitely is capable of  calculating very accurately the probability  that  one charged particle will affect another. This is Quantum Electrodynamics, again, you will have to go digging, Matt Strassler's Blog is excellent, on all levels.

Why is it considered the same fundamental force as electric attraction/repulsion when it follows wildly different rules? CAN it be derived from the rules of electric charges?

Magnetism  and electricity have to follow different rules, otherwise we would not be able to distinguish  between them, but they are connected and the rules, especially if you have the time to read about the Faraday tensor, ( and they have the same scalar and vector potentials behind them), more digging for you, sorry. But it's covered in the Faraday link above and  will show you they are not "wildly" different, just interestingly different.
One video that might interest you is Richard Feynmann being asked the same question as you are asking. Feynmann and Magnets. 
Best of luck with your research, which you will need to do if you really want to know more.
A: 
"But what force of attraction/repulsion, in that case, is a magnetic
  field representing?"

The magnetic field is representing part of the electromagnetic force.  Electric fields and magnetic fields are just parts of the same thing, and are intimately coupled.  Both are governed by photons as their mediated particles, and are intimately connected with light phenomenon.

"I understand a magnetic field is created when an electric charge
  moves, as well as by the particle spin of a charged particle. But is
  that all we've got on its origins?"

Pretty much.  Physics is just about the laws that govern the universe, not the why.  We have no test of why charge exists, or mass, or the nuclear strong and weak forces.

"Does it just show up when electric charges move and that's that?"

There is also a piece of fundamental information called "spin," which is a source of a magnetic dipole field.

"Why is it considered the same fundamental force as electric
  attraction/repulsion when it follows wildly different rules?"

It is a part of the picture, and if you look at Maxwell's equations, you will see that electricity and magnetism actually obey very similar rules.  The funny bit is actually that our universe doesn't have magnetic monopoles, but has plenty of electric monopoles (electrons, for example).  Just as you've learned that moving electric monopoles give rise to a new type of force called the magnetic force; moving magnetic monopoles would give rise to the electric force.  In more advanced courses, it is shown that the total force a particle experiences is the same in different inertial frames, but that the part assigned to "magnetic" and the part assigned to "electric" can trade off with each other.

"CAN it be derived from the rules of electric charges?"

They are one and the same.  Check out discussions of classical electromagnetism in the context of special relativity.  There is a nice little discussion at https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity that shows inertial frame transformations.
A: I am looking from the unified theory. I feel that all basic forces of Nature should have same origin. Gravitational , Electrostatic , Magnetic, Nuclear .
Let us put them on a force spectrum, one end may be Gravitational and other end may be nuclear forces.
I am trying to understand the basic unifying link between all these forces.
Like moving electron produces magnetic field, so does moving mass produces Gravitational field.
I don’t know , how far I am right ?
