Here's a video from a series I'm a fan of..
From here, the answer isn't really that interesting. QED is the fundamental theory for individual electromagnetic interactions. Magnetism is the result of a massive conglomeration of electromagnetic interactions.
A "magnetic field" is just a term for a specific subcomponent of the electromagnetic field. So a magnet is just the result of a bunch of QED interactions between electrons and photons on a massive scale that coincidently manages to cancel out the electric field components.
The details of how this cancellation is done is kinda like analyzing the individual grains of sand in a sandcastle (at least to me). I think a simple model of a pure electric field goes by the name "coherent" something something. I visualize it as a bunch of photons floating through air constantly going through a photon -> electron+positron -> photon cycle that manages to yield a net 0 magnetic field. A pure magnetic field would be similar. e.g. as interesting as counting sand grains.
But ultimately -- the actual interesting question is "what is the fundamental first principle cause of the electromagnetic interaction." And that would be what the physicist in the video I linked above goes into. The great Richard Feynman wrote a book about this topic with the aim of being readable by common folk. It might be of interest to you.
Personally, I think you would get great suggestions on /r/physics. But since you're here...
Since you seem like you're just dipping your toes in the water, you might want to start off with something basic like Hawking (A Brief History of Time, The Universe in a Nutshell).
I highly recommend Feynman's QED, it's short but there's really no other book like it. Anything else by Feynman is great too. I found this on Amazon and though I haven't read it, I can tell you that he was the greatest at explaining complex topics to a mass audience.
You'll probably want to read about relativity too, although my knowledge of books here is limited. Someone else can chime in, maybe. When I was a kid I read Einstein for Beginners and loved it, but that's a comic book so it might not be everyone's cup of tea.
If you really want to understand quantum mechanics and don't mind a little calculus (OK, a lot), try the textbook Introduction to Quantum Mechanics by Griffiths. Don't settle for hokey popular misconceptions of how QM works, this is the real thing and it will blow your mind.
Finally, the most recent popular physics book I read and really enjoyed was The Trouble with Physics by Smolin. It's ostensibly a book about how string theory is likely incorrect, but it also contains really great segments about the current state of particle physics and the standard model.
I'll recommend QED by Richard P. Feynman. It's not a textbook, and it has no math. Yet it quickly leads to a solid understanding of QM.
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https://www.amazon.com/QED-Strange-Theory-Light-Matter/dp/0691125759/ref=dp_ob_title_bk
There's a book about it. It's pretty good, though admittedly about 80% was over my head.
Richard Feynman's "Strange Theory of Light and Matter" explains why. It's all about probabilities.