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The magnetic field of an atom, say H (one electron revolving around a proton) is composed of two separate contributions: an orbital magnetic moment and a spin magnetic moment. We get magnetic moments because of electromagnetic induction associated with each type of electron motion. The electron revolving around the proton with a certain angular momentum is related to the associated orbital magnetic moment (and, yes, we model it as a wire loop with current expressed in terms of the orbital velocity). This can be calculated using elementary mechanics. The spin magnetic moment is a little trickier because it is caused by the intrinsic spin of the electron which is a quantum phenomenon; therefore, this spin magnetic moment can only be calculated using quantum mechanics. In the end, we get a certain net magnetic moment associated with this motion of the electron in an atom. It is important to notice that this atom acts as a magnetic dipole (remember, as of yet, magnetic monopoles don't exist!).
If we take a bulk magnet like you might have on your fridge, it would be composed of atoms. Each atom operates as a magnetic dipole, but due to internal atomic structures, the atoms actually align themselves into separate magnetic domains, each with a net magnetic moment (caused my individually adding up those atomic dipole moments). In a common magnet (ferromagnet), these domains all point the same direction and stay that way, creating a permanent bulk magnetic dipole.
So in a way, there is no difference between the magnetic field of an atom and that of a magnet because one is simply a sum of the other, making both magnetic dipoles. Of course, here we see that there's nothing that fundamental about the magnetic force. In reality, the magnetic force is a relativistic effect caused by the movement of a charged particle, making electricity (i.e. charge) the underlying property at work here. Electromagnetic induction is really a shortcut to understand the magnetic force without worrying about relativistic mechanics. If we were to discover magnetic monopoles (i.e. "magnetic" charge), this would re-write a lot that we know about electromagnetism because we couldn't label all magnetic forces as by-products of moving electric charge
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