electromagnetism - 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.