In general, the "Inspector of Plays" was a position in the British government introduced in the eighteenth century. In order to "protect" the public from the indecencies expected out of theatre, Britain mandated that all plays be submitted for government approval before being performed. The Inspector of Plays then, well, inspected the plays, and approved them or forbade them, often with certain parts censored.
The history of censorship in theatre, particularly British theatre, is incredibly long and fascinating, so I highly recommend looking for further reading. The Victoria and Albert Museum also just hosted a fantastic exhibit on theatrical censorship, so they may also have resources available for your research. I've included a few links below that might be helpful in your research.