Lenz's law (pronounced (IPA) [ˈlɛntsəz lɔ]) was formulated by German physicist Heinrich Lenz in 1834 and gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction.
For a current induced in a conductor, the current is in such a direction that its own magnetic field opposes the change that produced it.
Connection with law of conservation of energy 
Lenz's Law is one consequence of the principle of conservation of energy. To understand why, consider a permanent magnet that is moved towards the face of a closed loop of wire (eg. a coil or solenoid). An electric current is induced in the wire, because the electrons within it are subjected to an increasing magnetic field as the magnet gets closer and this produces an emf (Electromotive Force) that acts upon them. The direction of the induced current will depend on whether it is the north pole or the south pole of the magnet that is approaching: an approaching north pole will produce an anti-clockwise current (from the perspective of the magnet) while an approaching south pole will produce a clockwise current.
To understand the implications for conservation of energy, suppose that the above description was not true and that the induced currents were produced in the opposite directions to those described. Then, for example, the north pole of an approaching magnet would induce a south pole in the nearest face of the loop. The attractive force between them would accelerate the magnet's approach, and this would make the magnetic field increase more quickly. This would increase the current in the loop, which would produce a stronger induced magnetic field, a bigger force of attraction, yet more acceleration, and so on. Both the kinetic energy of the magnet and the rate of energy dissipation in the loop (due to Joule heating) would increase. A small energy input, to move the magnet forward, would produce a large energy output, which clearly violates the law of conservation of energy.
The above scenario is only one example of electromagnetic induction. Lenz's Law ensures that all induced currents have magnetic fields that oppose the change that induces them.