IPTG and lac operator with e coli for foreign gene question
The lac operon contains genes which are important for the metabolization of lactose as an energy source - normally glucose is used for this purpose. Usually the operon is tighly regulated and as long as there is another source of energy it is kept in an inhibited state.

The presence of lactose removes the lac repressor from the lac operon and allows the expression of the genes and thus allowing the metabolization of lactose. The mechanism can be turned on and off depending on the presence of lactose.

IPTG is a substance which mimicks the presence of allolactose (a metabolite of lactose) and it can activate transcription from the lac operon. As IPTG (in contrast to allolactose) cannot be hydrolyzed by β-galactosidase, it's concentration in the cell stays the same. Using the lac operon and IPTG enables you to switch on the expression of the gene on your plasmid and to start the overexpression.