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So why does a magnet attract another magnet?
A magnet works because these molecules often have an extra valence electron that is unbalanced. As it spins around the atom (this is not what it's doing, but the analogy works in this case) it creates a mini current-loop. Almost like a very small circular wire with a current running through.
All of these atoms arranged in series can become aligned in their spin of their valence electron, which is like a bunch of mini current-loops all aligned in the same direction. In practice they are not perfectly aligned, but statistically they are pointed in somewhat the same direction on average.
Adding all of these mini current loops together, you can think of the magnet as just one big current loop running around the outside, like a solenoid. Although magnetic field fall away from the source like 1/r^2, because with a solenoid you have the opposite side, where current is running the other direction, it actually falls off like 1/r^3 (at large distances that is). This is similar to an electric dipole (two opposite electric charges in close proximity). Because they are close, their electric fields are somewhat cancelling each other out, so the force on a distant object falls away with a cube of the distance, instead of a square of the distance (like 1/r^3 instead of 1/r^2). In fact the electric field for a dipole looks almost exactly like the field around a bar magnet, and they can be treated as mathematically the same. Sometimes a bar magnet is called a "magnetic dipole" for this reason.
The magnetic force is an 1/r^2 force of course, but it does not intrinsically attract, it's actually a force in the right-hand-rule direction to the velocity. In other words, it's a "turning" force that seeks to bend the charged object's trajectory. However, a closed current loop in any magnetic field can indeed feel a net force in a particular direction. That's what's going on here, treating the attracted magnet as another wire loop.
