We have magnetic fields all around our Earth. They are what make magnetism in the Earth. You know the Northern Lights you see in the sky that look like a firework show. Well that is actually our magnetic fields moving around in the sky. There you go. If that isn't enough look it up on Google. Hope I helped.
To put it into super simple words: You know that a magnet attracts a metal. The magnetic field is the area around a magnet where it has an effect on metals.
Not much is fully understood about magnetic fields. As far as how a magnet works, it's all to do with the magnetic component of an electromagnetic field and electromagnetic interaction. All substances contain things called domains. You can imagine these as little arrows, all arranged ina particular way for each substance. In most, the arrows point in all different ways and there is no order so the material is not magnetic. However, in Iron, Nickel and Cobalt, these domains all point in a particular direction, meaning the 'poles' are alligned and the material is magnetic. Some materials can have an induced magnetism, as in, when they are placed in a strong magnetic field the messy domains are all pulled to face in one direction, making the material magnetic. As for the orientation of the domains, that's to do with angular momentum and electron spin - pretty in-depth stuff. Oddly enough, I asked my Physics teacher about this yesterday, I found this website very useful: hyperphysics.phy-astr.gsu.edu/hbase/hframe.html