Magnetism is the component of the electromagnetic force that is generated by moving electrical charges. If you change your speed so that the charges appear to be stationary (imagine running along side them) then the force will appear to be electrical rather than magnetic. For an electromagnet (a wire with current running through it) you could potentially "run along" with the current and instead of seeing a current with a magnetic field you'd experience a charged wire with an electrical field. The relationship between electrical and magnetic forces and velocity is described by Maxwell's equations, and has been known since the 1800s. Because "electric" and "magnetic" fields are just two ways of looking at one thing, depending on your relative velocity, they are considered one, "electromagnetic," field.
On a microscopic/quantum mechanical level the electromagnetic field results from charged particles passing photons back and forth to each other. You usually experience photons as particles of light, which is electromagnetic radiation, but there are photon versions that are purely magnetic or purely electrical, and the version you get depends on your velocity. This is the quantum basis of Maxwell's equations. When particles with opposite charges swap photons they attract each other, and that includes moving charges (magnets). For magnets, instead of referring to positive and negative charge we generally refer to north and south poles, or up and down, but again, opposites attract.
For magnetic materials (fridge magnets, etc.) the magnetism comes from the atoms, nearly all of it from the electrons. The real (quantum mechanical) picture is a weird one, but the electrons behave in many ways *as though* they were little charged spinning balls orbiting around a nucleus like planets around the sun. If charge is spinning or orbiting it's moving, and voila! you have a magnetic field. In most materials all the electrons move in random ways and cancel each other out, so you don't notice the magnetic effect. In magnets though the material properties are just right to make all the orbits align in the same direction, and so they all add up and you say "look, a magnet!"
Electromagnetism is one of the four forces in nature (yep, only four). The force is carried by photons (AKA light particles), and acts on any particle with electrical charge. Charge can come in two flavors, which we call positive and negative. Gravity is another force, and it should be carried by particles called gravitons, which have not yet been detected but which we expect to exist. Just like electromagnetism, gravity has "charge," but it only has one flavor, and we call it "mass." Particles with electrical charge and no mass will ignore gravity but be affected by electromagnetic fields, and vice versa. (Ignoring general relativistic stuff, which is complicated.)
(In case you care: the other two forces are the "strong force" which holds the nuclei of atoms together, and the "weak force," which sometimes breaks them up. Each has it's own version of "charge" and a special particle that carries the force.)
