A Licence Manager is a peice of software that runs on a machine and has the job of handing out authorization tokens to programs. It is an outgrowth of the idea that a software
company should have control over how their software is used.
The idea is simple. When you buy the software, you buy a licence. You can pay different amounts based on what type of licence you need. Many companies will sell "Site Licences" (which mean different things to different vendors, but usually allow you to run as many copies as you want) or just buy a fixed number of licenses. You are then given a license file, which contains some cryptographic hashes (thus you can't just make one yourself). The licence manager then takes this file, and hands out keys to individual processes as it is asked for them, until you run out, at which point it refuses, and the software wont run (until someone else quits and thus hands back the license).
Many license managers provide network services. Thus many licenses allow you to install the program on as many workstations as you like, and still be constrained by the license. Often, the keys are tied to an individual host, so that if you wish to move the license manager to a new server, you must get a new license file.
This often results in what is known as License Manager Hell.