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Charged particles are coupled to the electromagnetic field. These means that they can change the shape of the field, and if the field is nonzero in their vicinity, they will experience some kind of force.

For electric charges, this picture is intuitive. Two electrons generate an outward-pointing electric field. If they're next to each other, each electron's field pushes the other one away, and they move apart.

Whenever a charged particle has some kind of motion associated with it, it generates loops of magnetic field around it. Similarly, whenever a charged particle has some kind of motion and is in a magnetic field, it will experience a force perpendicular to the direction of the magnetic field. The geometry is a lot more complex, so you can get things pushing on each other at odd angles, but in many cases the directionality of the loops and the forces cancel out, and you get a normal attractive force.

In everyday circumstances, most magnetic fields are associated specifically with electrons. There are three common kinds of motions of electrons. Their intrinsic spin, which generates a dipole field focused on the electron. Their orbit around atomic nuclei, which also generates a dipole field but focused on the center of the atom. And their motion through a wire (or through space, like in a thunderbolt), which generates circulating magnetic fields around it.

Most magnetism you see is some kind of interaction of these three types. For example, a bar magnetic picking up a paperclip. The inside of the bar magnet has a bunch of electrons' intrinsic magnetic fields lined up. The electrons in the paperclip feel this, experience a rotational force to line up their magnetic fields with the bar magnets' field, and then once they're lined up experience an attractive force toward the bar magnet. An electromagnet also generates a magnetic field, but using the bulk motion of electrons through the wire, which also allows it to pick stuff up.

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If electric current flows in a straight cable, it creates a magnetic field circling around it. If you bend the cable into a ring or a coil, then those little magnetic circles arrange to form electromagnet. What we take from this is that if electric charges are circling, there will be a magnet.

Now matte has lots of electron in it. Those things are electrically charged. But they also "spin". They are however infinitely small, so this spinning is a bit weird, but the same ideas apply. Hence they have a magnetic field by our previous observation; they are microscopic magnets. To get a macroscopic magnet, they need to be aligned so that their little magnets don't just cancel each other out.

In most matter, there are a lot of those little buggers, chaotically swirling around (metals, very hot "gas" [plasma]) or orbiting in atoms. The chaotic ones usually contribute nothing unless we create a current again or apply a magnet from the outside. They just point wherever they want.

Those in atoms follow some quantum rules, which tends to pair them in opposing directions; hence they cancel. They pair up, but sometimes one or more cannot find such a partner. This turns the entire atom into a slightly larger (but equally strong) magnet. Now most metals have those lonely electrons on the outside and they tend to join the swirling chaos, leaving their atom behind. But some atoms like iron or nickel have them a bit further on the inside where they stay. Thus those metals are effectively a lot of little magnets.

They are however still probably just pointing wherever they want. But if one gets them arranged properly in the same direction ("magnetizing"), then we finally got a magnet! This can be done in several ways, e.g. by an external magnet (might be electrical) forcing them to arrange; one usually then heats the piece of metal and cools it again, this will lock them in place so that they stay a magnet even if our external magnet is removed.