How do magnets work? It's a tough question, even inspiring an internet meme. At the very bottom of all this is the fundamental concept that a magnetic field is the movement of energy (charge). So at the level of particles, how can it be that a particle that is not moving can still have a magnetic charge, as is the case with protons and electrons? This is a caused by a property of every particle that we call 'spin'. What is spin and how is it expressed by particles? We don't know! All we know is that there is some property of particles that gives them tiny magnetic fields. A particle that is not moving, but rotating on itself, would still count as 'moving energy', and hence the term spin is used. However, fundamental particles are infinitely small, and therefore the idea breaks down. For now, until a more satisfactory understanding is developed, the best we can do is move forwards using this analogy of a rotating particle. Atoms can be magnetic when their electrons are organized in such a way that more of them are spinning in one direction than the other. Iron, for example, has 26 electrons: 11 spinning one way, and 15 the other. So the net pull is in the direction of the 15. Of course there are always exceptions. Manganese has 10 electrons spinning one way, and 15 the other, so the imbalance towards the 15 should be even stronger. Yet manganese is not magnetic, because of a phenomenon called Exchange Interaction. Essentially, for the question of how magnets work, the more satisfactory you aim to make your answer the deeper down the rabbit hole you go...
Bonus:
Some of the complexity of the subatomic world is rather imaginatively animated in this quick video by graphic designer Markos Kay.
-E
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