How do magnets work?

Questions that often come up are, "How do magnets work?", or, "Why is iron magnetic?", or, "What makes a magnet?", or, "What is the magnetic field made of?".
Those are good questions, and deserve a good answer.  However, did you know that there is a lot about magnets at the atomic level that isn't known yet?  Just like with most of the other basic forces we are familiar with, such as gravity, electricity, mechanics and heat, scientists start by trying to understand how they work, what they do, are there any formulas that can be made to describe (and thus predict) their behavior so we can begin to control them, and so on.

The work always starts by simple observation (that's the fancy word for playing around with the stuff!).  That's why it's so important to have some "hands-on" experience with magnets.  Have you taken two magnets and tried to push like poles together?  How far away do you start to feel the repulsion?  How does the force vary with the distance between them?  When the magnets are moved off-axis to each other (moving them to the side and not head on) what does it feel like?  Could you describe it like trying to push two tennis balls together?  When you flip one around, what changes?  What about moving one around the other in a circle?  Try these things!  That's how you learn!  Only when you play with (observe) them will you begin to understand how they work. This is the stuff great scientific pioneers did, like Faraday, Lenz, Gilbert, Henry and Fleming.

What we can find out this way, is some of the basics of magnetism, like:

    the north pole of the magnet points to the geomagnetic north pole (a south magnetic pole) located in Canada above the Arctic Circle.
    north poles repel north poles
    south poles repel south poles
    north poles attract south poles
    south poles attract north poles
    the force of attraction or repulsion varies inversely with the distance squared
    the strength of a magnet varies at different locations on the magnet
    magnets are strongest at their poles
    magnets strongly attract steel, iron, nickel, cobalt, gadolinium
    magnets slightly attract liquid oxygen and other materials
    magnets slightly repel water, carbon and boron
    and so on