Question

What produces a magnetic field?

Answer 1

You want examples of things that produce a magnetic field?
Or just that cause? What causes the magnetic field...?

Answer 2

In short, a magnet (of any sort) produces a magnetic field. Now why a magnet produces a magnetic field is a deep and hard question.

Answer 3

Electric currents have magnetic fields associated with them. Any moving charge produces a magnetic field. Magnets are composed of atoms of elements like iron and cobalt which have special orbits of electrons which produce relatively strong atomic magnetic fields. The atoms align to reinforce the magnetic fields of of the atoms producing a large easily detectable magnetic field.