Magnetic induction is the creation of a magnetic field by passing an electric current throug a conductable wire of some sort. The current creates a magnetic field that surrounds the wire. This was first proven around 1831 by Michael Farraday. Farraday passed an electric current through an insulated wire, then passed the wire over a compass. Normally, a compass is affected by the Earths magnetic field, but this compass was affected by the current passing through the wire. This is the principle of all electromagnets, and has far-ranging application in the industrial world.
If you increase the intensity of the electric current, likewise, you will create a stronger magnetic field. The magnetic field will sweep outwards, from the north pole to the south, in circular lines that are easily traced with iron filings. Super-strong electromagnets may be created by passing large amounts of current through their wires. There is also another relationship with the number of coils, or loops, wrapped around the magnetic core, as to the strength of the magnetic field generated.
Magnetic fields and electric currents are very deeply linked. If you have a current flowing through a straight wire, you'll find a magnetic field around it, from Ampere's law:
B = mu_0*I/r
B - magnetic flux density (Tesla)
mu_0 - permeability of free space
I - current
r - distance from the wire
(how would you know there was a magnetic field there? How could you measure it?)
And if you move a loop of wire through a magnetic field, you'll find a current induced in the wire (Faraday's law):
V = - v*B*l
where V is the induced electromotive force or voltage
l is the length of the wire
and v is the velocity of the wire moving through the field.
Notice that we're dealing with vectors here. You move the wire through the field at right angles to the direction of the field and to the flow of current (what happens if you do it at a smaller angle? How can you measure the voltage induced?).
This is all because of the magnetic field acting on the electrons in the wire, and the moving electrons in turn generating a magnetic field.