When charges move they form an electric current. We say that a current of 1A flows when a charge of 1C passes a point in 1s.
An electric current in turn creates a magnetic field, B, which can be detected by its effect on nearby magnets and moving electric charges.
A charge q which moves with velocity v in a region of magnetic field B feels a force, called the Lorentz Force,
F = qv×B where B is measured in Tesla (10,000 Gauss = 1 Tesla).
A long straight wire carrying a current I creates a magnetic field which consists of circular B lines perpendicular to the wire and centered on it. The field falls with distance r from the wire according to the formula B=μ0I/(4πr2) where μ0 is a constant called the permeability of free space with value 4π×10-7Tm/A.
Ampere's law tells us that the magnetic field flow round a closed loop in space is proportional to the net current flowing through any surface bounded by the loop. Mathematically ∫B°dl=μ0I where I is the current through a surface bounded by the loop (and the crazy circle is the best I can do for a vector dot product in html).
Faraday found that a changing magnetic flux, a change in the number of magnetic field lines through some surface, induced an electric field round the perimeter of the surface. Faraday's law can be written ∫E°dl = -d( ∫B°dA)/dt where the B field is integrated over the surface bounded by the loop over which the E field is integrated.
The sign of the resulting field is also given by Lenz's law which states that the Faraday field acts to make a current flow in such a way that it tries to keep the magnetic flux constant. It can be shown that this is equivalent to saying that the Faraday field points in such a way as to conserve energy.
James Clerk Maxwell showed that Ampere's law was incomplete as it stood and that there must be an additional term allowing a changing electric field to create a magnetic field, as well as a current doing so. He showed that with the addition of that term then the complete set of equations describing the electric and magnetic fields predicated the existence of waves of electromagnetic energy travelling at the speed c=3×108m/s. Heinrich Hertz detected these waves soon thereafter and we now know that light is simply an electromagnetic wave lying in a suitable range of frequencies.
The law of Biot and Savart
The total magnetic field B created by a loop of wire dl carrying a current I is given by
B = ∫μ0Idl×r/(4πr3) where r is vector from the point of the measurement to the line element dl.