The basis of Faraday's law

greatest contribution to physics was to show that a voltage, E, is generated by a coil of wire when the magnetic flux, , enclosed by it changes

E = N×d/dt volts
where N is the number of turns.
(Vector quantities express this more rigorously, but Faraday was even less of a maths fan than me.) The flux may change because

A nearby permanent magnet is moving about.
The coil rotates with respect to the magnetic field.
The coil is wound on a core whose effective permeabilty changes.
The coil is the secondary winding on a transformer where the primary current is changing.
In electric motors and generators you will usually have more than one of these causes at the same time. It doesn't matter what causes the change; the result is an induced voltage, and the faster the flux changes the greater the voltage.

The effect of coil current
OK, that's all clear enough, but there is one other reason for alteration to the flux: current flow in the coil. Hans Christian Oersted discovered that an electric current can produce a magnetic field. The more current you have the more flux you generate. That, too, is easy enough to grasp. What needs a firm intellectual grip is to appreciate that Faraday's Law does not stop operating just because you have current flowing in the coil. When the coil current varies then that will alter the flux and, says Faraday, if the flux changes then you get an induced voltage. This merry-go-round between current, flux and voltage lies at the heart of electromagnetism. Calculating the sequence of operation just described is quite easy.

The 'flip side to Faraday'
Now let's spin our fairground ride in the other direction. Instead of getting an induced voltage by putting in a current we'll put a voltage across the coil and see what happens. Normally, if you put several volts across any randomly arranged bit of wire then what will happen will be a flash and a bang; the current will follow Ohm's law and (unless the wire is very long and thin) there won't be enough resistance to prevent fireworks. It's a different story when the wire is wound into a coil. If the current increases then we get flux build up which induces a voltage of its own. The sign of this induced voltage is always such that the voltage will be positive if the current into the coil increases. We say that the induced voltage will oppose the externally applied voltage which made the current change (Lenz's law). This creates a limit to the rate of rise of the current and prevents (at least temporarily) the melt-down we get without coiling.
Calculating this 'flip side to Faraday' is also easy; we take his law in its differential form above and integrate:

= ( E.dt ) / N webers
where E is the externally applied voltage and N is the number of turns.
We'll call this the integral form of Faraday's law.
Inductor with AC applied
Let's apply a sinusoidal voltage, frequency f, RMS amplitude a -
E = (√2)a.sin(2.f.t)
Substituting this into the integral form of Faraday:
= ( (√2)a.sin(2.f.t) .dt ) / N
= ((√2)a/N) sin(2.f.t) .dt
= (-(√2)a/(2.f.N)) [cos(2..f.t)]0t
The expression with the limits of integration will always be between -1 and +1 so that the peak value of flux is given by
pk = (√2)a/(2.f.N)
pk = a / (4.44 f.N) Wb
Example: If 230 volts at 50 Hz is applied to an inductor having 200 turns then what is the peak value of magnetic flux?
pk = 230 / (4.44 × 50 × 200)
pk = 5.18 mWb
One important general point: if your winding has to cope with a given signal amplitude then the core flux is proportional to the inverse of the frequency. This means, for example, that mains transformers operating at 50 or 60 Hz are larger than transformers in switching supplies (capable of handling the same power) working at 50 kHz.
Using inductance
If the material permeability is constant then the relation between flux and current is linear and, by the definition of inductance :
= L×I/N amps
where L is the inductance of the coil.
Substituting into the integral form of the law:
L×I/N = ( E.dt ) / N
I = ( E.dt ) / L amps
If E is a constant then this formula for the current simplifies to:
I = E×t/L amps
Example: If a 820 mH coil has 2 volts applied then find the current at the end of three seconds.
I = 2×3/0.82 = 7.32 amps

Conclusion:
Satisfy yourself that this result above is consistent with our original formulation of Faraday's law: voltage is proportional to the rate of change of flux. Consider also how useful this integration method is in practical inductor design; if you know the number of turns on a winding and the voltage waveform on it then you integrate wrt time and voilá you have found the amount of flux. What's more is that you found it without knowing about the inductance or the core: its permeabilty, size, shape, or even whether there was a core at all.
Caveat (there's always one): The coils decribed here are idealised.