You have asked a very difficult question. It often takes many years of study to understand the relationship between magnetism and electricity. In order to really understand this relationship, it helps to understand some of Einstein's theory of relativity. Fortunately, we can understand most of electromagnetism without knowing much about relativity.

In brief, people often call the fields of electricity and magnetism by the term "electromagnetism." The reason for this name is that electricity and magnetism are very closely related. They both deal with the movement of electric charges like the electrons that flow through wires, that we know as electricity. Basically, the flow of electricity through a wire creates magnetic fields, and certain types of magnetic fields (they have to change with time) cause the flow of electricity. Generators of electricity use magnets to generate the electricity.

Electric motors use both electricity and magnets to create motion. The incoming electricity provides the power and generates magnetic fields which convert the power into motion. Motors are basically the opposite of generators. One uses electricity to generate motion and the other uses motion to make electricity.

Magnets are surrounded by a magnetic field, and an electrically charged particle, such as an electron, moving through a magnetic field, may feel a force, causing the particle's path to curve.

So where do magnetic fields come from? That's the answer to your second question. Moving charges create magnetic fields. What do you think are the moving charges inside of a magnet?

Motors are based on these facts about magnetism. An electrical current is just a bunch of moving electrical charges. Therefore if you place a wire carrying a current in a magnetic field, there will be a force on the wire, and the wire will move. By arranging everything in the right way, you can make a motor.