Lenz's Law

The induced current will flow in a direction so that the induced magnetic field they create opposes the action of the inducing field. That is, total energy is conserved.

Recall Magnetic Fields#Magnetic Field of a Coil or Solenoid

The south end of the magnet is moving into the coil.
The electrons in the coil want to oppose this and create a south pole on the left. To do this, the electrons will move out of the page at the top, and into the page at the bottom (for this coil, towards the right).

We can find this using right hand rule, with the thumb pointing to the right (north pole), and the fingers being the direction of the current.

The south end of the magnet is moving away from the coil.
The electrons in the coil want to oppose this and create a north pole on the left. To do this, the electrons will move into the page at the top, and out of the page at the bottom.

In this figure, the coil is backwards compared to the one in the example. We can still use the right hand rule to predict the current direction.

The created magnetic field opposes the rate of change of the original magnetic field, not the field itself (this should be evident given the previous explanations).