CASE 1: A magnet moving into a coil.
The animation below is a simulation about electromagnetic induction when a magnet is inserted into a coil. Click on the play button and make your observations.
When there is relative motion between the solenoid and the magnet, the solenoid cuts through the magnetic field lines (magnetic flux) which causes an induced electromotive force (emf). When there is no relative motion, the solenoid does not cut the magnetic flux hence no induced emf.
CASE 2: A coil inserted to enclose a magnet.
The animation below is a simulation about electromagnetic induction when a coil is inserted to enclose a magnet. Click on the play button and make your observations.