Great question, Richard! Magnetism is mysterious to most people because it can be extremely complicated. But it has simple origins. You’re right that unpaired d-shell electrons play an important role. But that’s effectively a single-atom view. The next level of complexity is to look at nearest-neighbor interactions.

What is the interaction between two atoms sitting close to one another? How do the intrinsic magnetic fields (spins) of the electrons interact, and how much do the electrons themselves overlap? The energy of that interaction is called the Exchange Energy and it’s written:
E = -J * S₁* S₂,
where S₁ and S₂ are the direction of the spins (intrinsic magnetic field) of unpaired electrons on the first and second atoms.

J is a coupling constant that can be positive or negative , and its value depends on an enormous number of factors including the energy level and atomic shell the electrons reside in (s, p, d, f), the crystal structure and specific crystal site, corrections due to Einstein’s theory of relativity (for larger atoms like Mo), just to name a few. For most materials, determining the value of J requires supercomputers running quantum mechanical simulations. I realize that is an unsatisfying answer. There are teams of researchers all over the world who specialize in computing J, among other things, and much of the time even they get an incorrect result.

Once J is known, we can say something about what type of magnetic order we expect from two atoms sitting next to each other, and then we can extrapolate to large systems of atoms that make up everyday objects like refrigerator magnets. Magnetic systems always try to minimize their energy . So if J is negative, S₁ will rotate around until it is equal to -S₂, so that E is a negative number.
E = -(negative)(negative)(positive) = (negative).
In other words, a material with negative J will be anti-ferromagnetic. The spins will be anti-parallel to one another, and since all the internal magnetic fields cancel out, there is no external magnetic field produced by anti-ferromagnets. They won’t stick to your fridge.

On the other hand, a positive value of J will tend to cause S₁ to rotate to be parallel S₂, so that E is still minimized.
E = -(positive)(positive)(positive) = (negative).
These materials have ferromagnetic ordering if the value of J is high enough. Ferromagnets produce an external field and some are strong enough to stick to your fridge.

Thermal energy can easily wash out the effects of the exchange energy if J is small. That’s the difference between a paramagnet and and a ferromagnet. A ferromagnet has a J high enough to keep all the spins in a chunk of iron (or whatever) pointed the same way, even though temperature is always trying to randomize the direction of the spins. A paramagnet has a small J, but can be temporarily magnetized by applying an external magnetic field (for example, in a coil of wire with electrical current running through) because paramagnets still have unpaired electrons. With the assistance of an applied field, some paramagnets would stick to your fridge too. Most paramagnets become ferromagnets if you cool them down enough and remove that randomizing thermal energy.

That gives us enough to talk about Cr and Mo . Cr is a small atom and is stable in the bcc (body centered cubic) crystal structure. The unpaired electrons are in the 3rd d-shell (3rd row of the periodic table). It has a strong, negative exchange constant J, giving antiferromagnetic order.

Mo is a larger atom but also crystallizes as bcc. It has the same number of unpaired electrons but they are in the 4th d-shell, further out from the nucleus. That results in a very different overlap between electrons on neighboring atoms. For Mo, J ends up being small and positive, giving paramagnetic order.

Also, as a side note, there are many more magnetic materials than just the magnetic elements. I work with the Heusler compounds, which originally gained fame as the first ferromagnetic compounds composed entirely of non-magnetic elements. The first one was Cu₂MnAl (diamagnet, paramagnet, paramagnet), which has stronger ferromagnetism than Ni metal. In Heuslers, the ferromagnetism (or anti ferromagnetism!) arises because unpaired electrons actually get traded between neighboring, dissimilar atoms. To do that without violating the Pauli exclusion principle (you can’t put two electrons with the same spi

Magnetism relies on the material having a net magnetic moment, a quantity that determines the amount of torque (how much it spins around) the material experiences in a magnetic field. The net magnetic moment for a bulk material, e.g. a cubic inch of iron, requires the alignment, in one direction, of small magnetic dipole moments. Magnetic moment on a fundamental level exists intrinsically (without any external stimulus) in elementary particles, such as electrons and particles that make up protons and neutrons, because these particles have spin -- a form of angular momentum. When these intrinsic magnetic moments are not somehow balanced out, they can align, resulting in a net magnetic dipole moment in the bulk material, with or without the help of an external field. What helps the alignment of small magnetic moments into a large collective one includes the number of electrons in a material, how they pair, AND the crystalline structure and microstructures. In fact, if we look closely at iron, we see that for iron to become independently magnetic, it needs to cool from high temperatures to allow a change in domain structure, meaning that the electron orientation of one domain (local area) in the iron aligns with other domains so that a net magnetic moment results.

The structural component of magnetism also means that the material itself does not need to be ferromagnetic. A really good example of this is the compound CrO², which is a ferromagnetic but neither Cr nor O is ferromagnetic alone. In short, alignment of small magnetic moments is an essential part of magnetism that cannot be entirely included in electron pairing. The alignment can also help explain why some metals don't respond to external magnetic fields as readily as iron - the small magnetic moments are not as easy to align because of electron pairing, crystal structures, domains with different orientations, and so on.

Great question! You are absolutely right, that unpaired electrons in the d shell do not explain why Fe, Co and Ni are ferromagnetic, Mn and Cr are antiferromagnetic, and other transition metals are nonmagnetic. If we imagine an isolated atom that is not part of an extended solid (for example, a single atom in a vacuum, or even a molecule that contains a single magnetic atom in it), then we only need to consider the presence (or absence) of unpaired electrons to know whether or not it is magnetic. Any atom with unpaired electrons is magnetic. There are all sorts of magnetic molecules that contain isolated metals with unpaired electrons with all sorts of types of atoms: even an oxygen molecule, O², is magnetic!

When you put a bunch of atoms together into an extended metal, however, the physics changes a lot. In metals, the valence electrons of each atom join together to form a sea of electrons that is shared across the whole metal. Electrons are no longer associated with a single atom, and so the concept of an individual atom's electron configuration is meaningless for an extended metal... we can't just count electrons and orbitals to see if there are any unpaired electrons like we can for an isolated atom! This is why you have been unable to understand metallic magnets using the concept of unpaired electrons.

The reason that some metals are magnetic while others are not is actually very complicated—so complicated that scientists are still trying to fully understand this question! However, we do understand some basic principles about magnetism in metals.

For example:
1. Magnetism is more likely in the middle of a row of the periodic table.
2. Magnetism is more likely at the top of a block of the periodic table.

In the case of pure elemental metals, this results in the only magnets being Fe, Co, Ni (which are at the middle of the top of the d-block) and Gd (which is at the middle of the top of the f-block). However, there are lots of other magnetic metals that are compounds with more than one element in it. For example, the compound MnCu2Al is a famous example of a ferromagnet. In this compound, the Mn is actually the magnetic element! In compounds like this, the above rules still apply. There are lots of magnetic metallic compounds based on Mn, Fe, Ni, and Co; some based on V, Cr, Mo, Ru, Rh, Ni and Cu; and relatively few based on other transition metals.

If you are interested in exploring some metallic ferromagnetic compounds, you can check out magnets.mrl.ucsb.edu, which contains interactive applications for exploring data about this type of compound.

Try, for example:
here

To see a plot showing different magnetic compounds based on different transition metals.