Electric forces are attractive or repulsive forces between "charged objects", e.g. comb and dry hair after some friction. Charged objects are those that carry some nonzero electric charge $Q$. The lightest – and therefore easiest to move – charged particle is the electron so the surplus or deficit of electrons is the most typical reason why some objects are charged.
Magnetic forces are attractive or repulsive forces between magnets, like magnetized pieces of iron. The amount of "magnetic dipole" carried by a magnet is completely independent of its electric charge. They're as independent as the gravitational and electrostatic forces i.e. as independent as the mass and the charge of an object.
For centuries, these two forces were thought of as independent. Only a few centuries ago, due to Faraday and others, relationships between the electric and magnetic forces began to be uncovered. Magnets may be produced by coils – by electric charges moving in loops. They become indistinguishable from bar magnets. Similarly, moving magnets produce electric fields.
In the middle of the 19th century, because of these "mutual influences" between electricity and magnetism, a unified theory was gradually found. Because electricity and magnetism influence each other, we need to talk about a whole – electromagnetism or, to point out that magnetism is related to moving electric charges, electrodynamics (dynamics sort of means "motion" or "reasons for motion").
James Clerk Maxwell wrote the unified equations for electricity and magnetism which exhibited a near perfect symmetry between electricity and magnetism. They are two independent "siblings" but they affect one another and the inner mechanisms in them are analogous. Maxwell's theory also implied that there are electromagnetic waves – disturbances in space where the electric field goes up and down and so does the magnetic field which is excited by the electric one and vice versa. Moreover, he proved that light was a special example of the electromagnetic wave.
In the 20th century, it was realized that the existence of the other force follows from one force (e.g. magnetism followed from electricity) due to a symmetry between inertial observers who are moving relatively to each other, i.e. due to the Lorentz symmetry which underlies Einstein's special relativity. It was also found out that the electromagnetic waves may be thought of as collections of photons and that the exchange of the photon is the "reason" behind electric as well as magnetic forces.
So the photons are the messengers of electromagnetism – both electricity and magnetism. Electrons are the most important carriers of the electric charge which means that they're the most important particles that produce the electric and magnetic (when electrons are moving or spinning) fields. These fields arise and affect other pieces of matter (especially electrons) due to the "messenger role" of the photons. Photons are "units" of the electromagnetic waves.