Principle that dictates when current flowing through one coil of wire will induce an electromagnetic field (or EMF) and current in another nearby coil.
The Basics
When the current flowing through the first coil of wire is steady, an electromagnetic field will be produced in the second coil but NO current will be produced. Why? According to Faraday's Law, the magnetic field must change in order to produce a current.
If the current in the first coil is changing (circuit is opening or closing, another electromagnetic field passing through) the second coil will produce both an EMF and a current. Lenz's Law predicts that this current will oppose the change in the field of the first coil because coils are reactionary.
Applications
Mutual inductance is most commonly seen in transformers. Transformers are made of two coils of wire wrapped around an iron core. Since most houses are hooked up to alternating current or AC power sources, an AC current flowing through the first coil would induce a current in the second coil. Why is this useful? The number of turns the second coil makes is directly proportional to the current that will be induced- in other words, if both coils have the same number of turns, the current will be the same; if the second coil has half as many turns as the first, the current induced will be half as much as that flowing through the first. The standard 120 volt AC power source can be reduced to power devices that require a smaller voltage.
Some electronic devices run off direct current or DC power, so in standard transformers (those black boxy-looking things at the end of the cord instead of a normal plug) it is necessary to add diodes, which serve as a rectifier to convert AC power to DC.
Another more spectacular application of mutual inductance is the electromagnetic pulse or EMP. EMP is generally associated with nuclear weapons but devices that produce only an EMP and skip the wholesale devastation have also been developed. Basically, an EMP device (such as a flux compression generator or one of those nifty grenade-sized ones the Russians have been playing with) throws out a huge pulse of energy along the electromagnetic spectrum, and because of the principle of mutual inductance the pulse is absorbed into every circuit for yards (or miles) around, effectively killing all the electronics in the area. Who says being a nerd isn't useful?