A magnetic field is one of the fields used to track forces. It gives us a recipe to describe magnetic forces around the source of the field.
Gravitational fields are used to describe the orbits of planets around the Sun. Electric fields describe how currents flow in the power grid and how radios work. A magnetic field is how we keep track of the magnetic force created by many moving particles (or electrical currents). There are two nuclear fields as well. Fields are not simple concepts. They do not follow our usual experiences. A concept like Newton’s law of motion, that force is equal to mass times acceleration, can be seen every day. But even though they are invisible, fields are as real as the forces they allow us to calculate.
Magnetic fields come from electrical currents, whether in the Sun or a magnet. The strength of magnetic forces can be greater than the force of gravity (that’s why magnets work) but it is weaker than the electric forces between charges. It also works when the source of the magnetic field is electrically neutral!
What is a magnetic field? A magnetic field is the region of space near a body where magnetic forces due to the body can be detected. It’s the reaction of the particles that counts, not the region.
Edited 08/05/2014 to fix the bar magnet picture.