Why are some materials diamagnetic, others paramagnetic, and others ferromagnetic? 
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*Why are some materials diamagnetic, others paramagnetic, and others ferromagnetic?

*Or, put another way, which of their atomic properties determines which of the three forms of magnetism (if at all) the compound will take?

*Is paramagnetism to ferromagnetism a continuous spectrum, or is there no grey zone in between?
 A: There are a few decent rules of thumb for para- and diamagnetism.
A system is paramagnetic if it has a net magnetic moment because it has electrons of like (parallel) spins. These are often called triplet (or higher) states. In atoms and molecules, they occur when the highest occupied atomic/molecular orbital is not full (degeneracy > 2 * # of valence electrons). In this case, Hund's rules suggest that the electrons lower their energy by aligning their spins.
In contrast, a diamagnet has no magnetic moment because all electrons are paired.
Nearly all free atoms are paramagnetic because nearly all atoms have unpaired spins. The exceptions are the the last column of the s, p, d, and f block (2, 12, and 18). (Any that I'm missing?) For instance, that's an important property for the Stern-Gerlach experiments and magnetic trapping.
Most molecules, however, have fully paired spins. First off, most molecules have an even number of spins, except for free radicals, which are relatively unstable. To figure out if the molecule has a net magnetic moment (paramagnetic) or not (diamagnetic), you need to look at its molecular orbitals. The classical example is oxygen, which has a half-full (or half-empty) $\pi_{2p}^\ast$ orbitals, and nitrogen, which has a full $\pi_{2p}^\ast$ orbital. See: http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm. 
For crystals and solid-state materials, the question is more challenging, but it ends up coming down to the same question: is there a net magnetic moment because of unpaired spins, in which case it's a paramagnetic? or is there no net magnetic moment because all spins are paired, in which case it's a diamagnet?
Of course, in solid-state, there is a third situation, a ferromagnet. This is rather difficult to predict in real systems and is a major field of research. Some model systems (model system: a much simplefr mathematical model of a system) are solvable and give hints of what to look for. For instance, free spins in a lattice create a paramagnet by the argument above: the crystal has a net magnetic moment. You expect, in a magnetic field, the spin of one electron creates a magnetic field that can effect its neighbors. Since the system is paramagnetic, you might expect that the neighbors align with their local magnetic field, which is induced by their neighbors, and the whole crystal polarizes itself, creating a ferromagnet. This explanation is a mean-field Ising model. It gives a good intuition even though it's too simple to describe any real system.
