Flux is the number of magnetic field lines passing through a surface.  As you can see in the diagram, this depends on both the strength of the magnetic field (stronger field = more lines = larger flux) and the size of the surface (larger surface = more lines = larger flux).

When the flux through a surface changes, an electromotive force (emf - and not really a vector force as we know it) is induced.  This induced emf causes induced current.

As long as the magnetic field is changing, a current is induced. The induced current is always in such a direction as to keep the magnetic field constant.  The emf induced in a circuit is directly proportional to the time rate of change of magnetic flux through the circuit. Basically this means that the induced emf is directly related to magnetic flux as a function of time.
 

Lenz's Law states that the polarity of an induced emf will produce a current that creates a magnetic flux that opposes the change in magnetic flux through the loop formed by the circuit.  The induced current opposes any change in magnetic flux.

That is to say, the induced current must be in a direction so that the flux it produces opposes the change in the external (applied) magnetic flux. For example, if a bar moves to the right, then the induced current would be counterclockwise so that the force produced is directed towards the left.

There.  A brief overview of flux and induced current.  Move on to Project, Part A
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