Well even though humans have known about magnets for thousands of years, the real reason they work is still somewhat mysterious. Depending on how deep you want to investigate, it can be a very complicated phenomenon.
A magnet is, at its foundation, caused by the movement of electrical charge (yes this is the Biot-Savart Law). When charges (like electrons for example) move, they produce magnetic fields. If a charge moves in a loop, it creates a magnetic field. The magnetic field can be basically represented by an invisible set of arrows in space, that point in a direction. In the case of the loop, the arrows of the magnetic field will point out of the loop for a positive charge moving counter clockwise (this is the so-called "right hand rule"), and into the loop for a charge moving clockwise (some pictures are in the reference attached).
The magnetic field has a "north" end and a "south end". The little arrows from our moving charge are pointing in the "north" magnetic direction, and away from the "south" magnetic direction. It is a mysterious and interesting property of electromagnetism that two objects with fields that oppose each other, i.e., the little arrows of the field do not line up, so either the north end of one field is facing the north end of the other field, or the two south ends are facing each other, repel, or resist movement of the two objects towards each other with some force.
On the other hand, if the little arrows line up, the two objects attract each other with some force.
The little arrows are aligned along the so-called "magnetic lines of force".
The forces that result from magnetic fields (and for that matter, electric fields) are fundamental forces, and part of the four fundamental forces of the universe (strong, weak, electromagnetic and gravity). It is believed that these forces are caused, at their base, by particles passing "messenger particles" back and forth. In the case of electromagnetism, the messenger particle is a photon. How this actually works is still somewhat poorly understood, and is involved with the world of quantum mechanics, which is very counterintuitive but interesting.
However, what is important is that we understand it well enough to make models of magnetism, and to make predictions based on those models which we can verify and use. All of electricity and electric motors are based on our ability to design systems with magnets and to be able to build and use them.