The lac operon of E. coli is the classic example for describing inducible prokaryotic gene expression. One excellent video description of it can be found here.
The jist is this. Not all genes are turned on all the time. There are ones which are needed constantly, others which are only needed in specific types of cells, and then others which are ‘turned on’ in specific situations. It is on this last point which I will focus.
In order for a gene to be ‘turned on’, it must be ‘off’ in the first place. All this means is that an organism’s (relevant) DNA is not being transcribed, thus preventing translation and the manifestation of proteins. The way this occurs in E. coli by means of the lac operon is that the lac repressor is bound to a DNA sequence.
A repressor is itself a protein. It binds to an organism’s DNA, thus preventing RNA polymerase from transcribing anything. This is a physical blockade; the repressor prevents the RNA polymerase from physically attaching and running along a specific sequence of DNA. This is the default position for an inducible repressor.
The way the repressor is removed is simple to understand. It has a specific shape to it which enables it to bind to the DNA sequence. However, this shape can be changed if lactose is present. The lactose will bind to the repressor, thus causing an allosteric change in shape. This means the repressor is no longer the specific shape needed to attach to the DNA, so it releases its ‘grip’.
This release allows the RNA polymerase to continue with transcription. This, eventually, turns to translation. In this stage, enzymes are created, two of which are β-galactoside permease and β-galactosidase (there is a third which can be ignored here). The former of the two is membrane-bound. This means it becomes embedded in the cell membrane. This quickens the transport of lactose from outside to inside the cell. Think of it like a tunnel through which only specific shapes can fit.
Once these specific shapes (lactose molecules) pass into the cell, ß-galactosidase breaks them into their constituents, one of which is glucose. This is used as a key source of energy in many organisms, including E. coli.
Once concentration falls, lactose molecules are no longer bound to the repressor, making it free to resume its normal duties attached to DNA.