A change in an electric field generates a magnetic field. Anything which produces an electric field, which electrons do, if they move, move the electric field, thereby creating a magnetic field.

Now, imagine if you have, say, a rod with an negative electric charge on one end and a positive electric charge on the other. This creates what is called an electric dipole. If you have two of these rods, they will attach to each-other with the negative end of one rod to the positive end of the other, but the positive ends and negative ends will repel, just as you learned.

When you create a static magnet with magnetic domains, the same thing basically happens: every magnet has a north pole and a south pole. Opposites attract, likes repel, and you can test this yourself if you have two magnets. If you attach a magnet to a piece of non-magnetized iron, the magnetic field generated by the magnet will temporarily magnetize the iron in such a way that a pole opposite that of the magnet touching the iron will form next to the magnet's pole. When you take the magnet away, the iron de-magnetizes. This is done via magnetic domains that you heard about - how those form is understood by physicists.

The magnetic field lines indicate the direction of the magnetic field, so arrows coming out indicate a magnetic north pole, arrows going in indicate a magnetic south pole. We draw them with arrows just to show that there is a distinction between them, but really the arrows themselves are meaningless, just a convention to help us visualize it. The density of lines, the closeness they are together, indicates the strength of the field, and the direction indicates its polarity.