Static magnetic multipoles: this one is a little more complicated because it's not described by any of Maxwell's equations, at least not directly.
Let me start with an analogy. Hopefully you know that a charged object produces an electric field. But you don't have to have a net charge to produce a field. If you take a positive charge and a negative charge of equal magnitude and put them very close to each other, you'll still get an electric field, because the field from the positive charge and the field from the negative charge don't exactly cancel each other out. This is an example of an electric dipole. You can think of this as a "secondary source" of the field, which depends not on the total amount of charge, but on how the charge is distributed within the object.
Normally, when the total amount of charge is nonzero, the distribution of the charge has a small enough effect that we don't have to care about it, but when there is no net charge, the way the charge is distributed becomes important. Obviously, in order to do physics we need to have a physical quantity that describes the distribution. This is the electric dipole moment.
In fact, we can measure the electric dipole moment of an object and use it to do useful calculations even if we don't know anything about the charge distribution - or even if there may not be a charge distribution at all. In other words, one could imagine that there might be some unknown physical mechanism, completely separate from electric charge, that causes some object to have an electric dipole moment. So it makes sense to define an "electric dipole" as "something that has a nonzero electric dipole moment," whether or not that thing has a charge distribution.
The same thing applies to magnetic dipoles and the magnetic dipole moment. It works just like the electric dipole moment, except with the magnetic field and "magnetic charge" instead of electric field and electric charge. The thing is, as far as we know, there are no magnetically charged objects (the so-called "magnetic monopoles"). So the magnetic dipole moment never gets masked by magnetic charge, the way the electric dipole moment usually does.
As with the electric dipole, a magnetic dipole of any sort will generate a magnetic field. One kind of magnetic dipole is a small loop of current. If the current is made of physical charges moving around in a circle, then it will have some angular momentum. So once it was discovered that the electron has intrinsic angular momentum (spin), physicists naturally wondered whether that angular momentum was due to constituent particles moving in circles inside the electron. One way to test this theory would be to measure the magnetic dipole moment of the electron and calculate whether it corresponds to the prediction of the current-loop model. As it turns out, it doesn't. So evidently something else is going on; the magnetic dipole moment of the electron is not just produced by classical charges moving in a circle. It's something intrinsic to the electron. (Quantum electrodynamics correctly predicts the exact value of the electron's magnetic dipole moment, but it doesn't offer a simple physical picture.)
Best Answer
An electron is not a spinning ball of charge and the intrinsic spin of particles cannot be understood in such terms. Not only is it difficult to make sense of what it means for a pointlike particle to spin, but also when treating the electron as a spinning ball of charge one finds a value of the ratio between the magnetic moment and the angular momentum that is a factor $2$ too small.
To understand why a rotating charged ball generates a magnetic field, note that every charge on the ball will move in a circle, so there is in fact a current, and that current will generate a magnetic field.