Defining magnetic flux precisely
Take a flat loop of area A sitting in a uniform magnetic field B. If the field points straight through the loop (perpendicular to its plane), the flux is simply \Phi_B = BA. If instead the field is tilted, only the part of the field aimed through the loop counts. We measure the tilt with the angle \theta between the field and the loop's normal (the line sticking straight out of its face).
Magnetic flux. Units: the weber (Wb = T·m²). Only the field component perpendicular to the loop's area contributes.
Faraday's law: the rate of change
Now the law itself. Faraday's law of induction says the induced EMF equals the rate at which the flux changes. If the loop is actually a coil of N turns wound together, each turn sees the same changing flux and their EMFs add, so we multiply by N.
Faraday's law. The EMF (in volts) is N times the rate of change of flux. The minus sign encodes direction — Lenz's law, next guide.
The EMF \varepsilon is a push measured in volts — the same kind of push a battery gives, except here it is created by changing magnetism rather than chemistry. For the size of the effect we use |\varepsilon| = N\,|\Delta\Phi_B / \Delta t|; we sort out the sign (which way the current flows) separately in the next guide.
What the flux really counts
Two subtleties reward attention. First, flux cares only about the field through the loop; a field lying in the loop's plane contributes nothing. Second, the true law uses the instantaneous derivative d\Phi_B/dt; the \Delta\Phi/\Delta t we used above is an average over the interval. When the field changes smoothly and steadily, as in the example, the two agree exactly.