When is Lac operon turned on and When is it turned off?

It depends on presence of lactose and Glucose in bacterial environment. The operon is turned on when available food source of bacteria is only lactose.
Introduction :
this operons codes for 3 enzymes involved in the catabolism of lactose sugar . lac Z codes for beta Galactosidase which hydrolizes lactose to Glucose and Galactose , lac Y codes for permease which facilitates the movement of Galactose into the cell , lac A gene whose function is not known.
The regulatory portion of this operons consists of 3 sites and 1 repressor gene . the first site is CAP binding site , the second is promoter site , the third is operator site . the repressor gene lacl codes for a repressor protein .
Turned off Completely
when Glucose is the only sugar available or no Glucose nor lactose or large amount Glucose with lactose
the repressor gene produces repressor protein which binds to the operator site which is downstream of promoter site and this blocks progress of RNA polymerase so no transcription from genes of metabolic pathway (structural genes). Because Glucose is available adenyl cyclase is inhibited so no cAMP is available to combine with CAP protein and so no active CAP so cannot bind CAP binding .
Turned on Completely
there is no Glucose but only lactose present
small amount of lactose is converted into allo lactose that binds to repressor protein and changes its conformation so it cannot bind to operator site . because no Glucose is available adenyl cyclase is available and produces cAMP which also binds to CAP protein to form active CAP which binds to CAP binding site. This allows RNA polymerase to effectively initiate transcription continously.
Turned on not completely
When Glucose (little amount ) with lactose :
little cAMP is produced and cannot bind CAP protein to Form active CAP , while Allolactose will bind repressor protein to prevent it from binding to operator.