The electromagnetic force, also called the Lorentz force, explains how both moving and stationary charged particles interact. It's called the electromagnetic force because it includes the formerly distinct electric force and the magnetic force; magnetic forces and electric forces are really the same fundamental force. The electromagnetic force is one of the four fundamental forces.
The electric force acts between all charged particles, whether or not they're moving. The magnetic force acts between moving charged particles. This means that every charged particle gives off an electric field, whether or not it's moving. Moving charged particles (like those in electric current) give off magnetic fields. Einstein developed his theory of relativity from the idea that if the observer moves with the charged particles, magnetic fields transform into electric fields and vice versa! One special case of the electromagnetic force, when all the charges are point charges (or can be broken up into point charges), is Coulomb's law.
Because calculating the force from every single individual charge on every other individual charge is ridiculously complicated, physicists have developed tools to simplify these calculations. These simplified calculations turn into the macroscopic, everyday phenomena listed below:
Modern physics has unified the electromagnetic and weak forces into the electroweak force. A full understanding of the electromagnetic force and the full implications of electromagnetism takes many years of study. Some good places to go for more information on electromagnetism include hyperphysics.
Below is the Scishow's series on fundamental forces part 4a (electricity) and 4b (magnetism):
And here is part 2.
For further information please see the related pages below: