At the atomic, or quantum, level, what is it exactly that causes magnetism ?

A moving electrical charge (electric current) produces a magnetic field — macroscopic flow in wires, for example.

Magnetism is caused by electrons orbiting the nucleus. However, the major contributor to magnetism is the inherent magnetic property of electrons that we call, somewhat misleadingly, "electron spin"... but they are not actually spinning but a quantum term applied to them.

Every electron is a tiny magnet due to its inherent magnetism (what we call electron spin). The alignment of the electron spins makes a lump of iron into a magnet.

All atoms have electrons with electron spin and magnetic fields due to their orbits about the nucleus. But not all materials are magnetic. If the electron spins of an atom's electrons are aligned oppositely, their magnetic fields cancel. That's what happens with aluminium, paper, water and other non-magnetic substances.

Each iron atom, on the other hand, has four electrons whose spin magnetism doesn't cancel. They line up. Aligned magnetic fields make matter magnetic.

In iron the aligned-field electrons spontaneously couple and form small long-lasting domains. The spins inside these microscopic domains are almost perfectly aligned. Most domains, though, aren't aligned. In common un-magnetized iron, many domains are randomly oriented.

Simply bringing a magnet (and its magnetic field) near iron will align its domains or cause those aligned with the external field to grow at the expense of their neighbors. With a stethoscope, we can hear little domains click into position! This akignment of the domains produces an overall net magnetism.

When we take the magnet away from pure iron, its magnetism goes away shortly because thermal motion of its atoms jiggles the domains out of alignment. On the other hand, we can make permanent magnets out of many special iron alloys that are different in composition. Once the domains of these iron alloys are aligned, they are much harder to randomize.

Electrons orbiting the atom act effectively create a tiny electric current. According to Ampère's law, an electric current produces a magnetic field. The electrons also have a quantum property called spin which can be involved in generating a magnetic field.

From Wikipedia: The origin of the magnetic moments that create the magnetisation can be either microscopic electric currents corresponding to the motion of electrons in atoms, or the spin of the electrons.

There are three magnetic fields in an atom. The nucleus, due its spinning motion, generates a small magnetic force. Electrons also generate a magnetic field from spinning. The resultant magnetic field (using vector addition of the nucleus and electron fields) is the third field in an atom. This resultant magnetic field is the overall magnetic field of the atom.

Basically, moving and spinning of charged particles (electrons, protons etc.) cause magnetism.

Electron spin causes magnetism. Although electrons are discribed as wave clouds, these can be described as rotations. The interactions of these spins with neighbouring electrons from different atoms are responsible for the different types of magnetism (ferromagnetism, the most well known).

Electrons do not spin. They are point particles.

Magentism is, in fact, a relativistic effect. Its origin is the electrostatic force between charged particles, corrected for the movement of those particles using special relativity. This is a direct effect for electromagentism (it is special relativity applied to the drift velocity of charge carriers).

Spin arises from including special relativity in the Schroedinger wave equation - it is actually a misnomer. The resulting equation is called the Dirac equation. Permanent magnetism arises from alignment of spin magentic moment of nuclei. But once again, special relativity is the source of the moment.