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.