Statement. Lenz's law, formulated by the German physicist Heinrich Lenz, gives the DIRECTION of an induced current (which Faraday's laws alone do not specify): the direction of the induced current is always such that it opposes the cause responsible for its production. If a changing flux is the cause and an induced current the effect, the induced current always acts, through the magnetic field it itself creates, to oppose that very flux change.
In the emf formula. Incorporating Lenz's law into Faraday's law supplies the standard minus sign, ε=−d(NΦB)/dt -- not a mathematical formality, but a literal encoding of the physical statement that the induced emf's polarity always opposes the flux change producing it.
Application technique. To find a direction: (i) determine whether the relevant flux is increasing or decreasing; (ii) the induced current flows in whichever sense creates its OWN magnetic field that opposes that particular change (reinforcing a decreasing flux, or counteracting an increasing one); (iii) apply the right-hand rule to that self-field to read off the current's actual direction around the loop. Equivalently, for a magnet approaching/receding a coil, the coil's near end becomes whichever pole (N or S) repels an approaching magnet or attracts a receding one.
Proof from energy conservation. If a magnet is moved towards/away from a coil, Lenz's law says the induced current opposes this motion, so external work must be done against a resisting force -- this work converts to electrical, then thermal, energy, consistent with energy conservation. If (hypothetically) the induced current instead HELPED the motion, a magnet nudged towards a coil would accelerate in with no external energy input -- a perpetual motion machine, which is never observed; since this contradiction is impossible, Lenz's law must hold, proving it is a direct consequence of energy conservation rather than an independent postulate.