Multiple sites served from single site

I am trying to be succinct, but I find this a hard problem to describe. Probably why my searches turned up nothing! All I want for an answer is the term to research, a name for what I am trying to do. Or to be told it's daft in the first place!

I code for a community site using MySQL, PHP, javascript etc. I am being asked to support the franchising of that site to other communities.

Each community would have it's own domain name but point to the same server. The server should respond by providing the site for that url.

Those sites would share some of the same code and database, but prioritise and sometimes omit different content. Similar areas sometimes even need to be known by different terms (for cultural or language differences).

For example, suppose we had a London community and a Manchester community. The London site would offer information on the underground train system, whereas Manchester would not. Both would offer information on restaurants.

Is this good practice? I have tried searching for methodologies or best practices but haven't found anything like this being mentioned.

Alternatively I suppose I'd have to create separate sites. In this case I wonder how to maintain sites in this manner. Multiple sites with overlapping code with differences.

I can see how some of this could be stored in the database, but that approach seems unwieldy.

So basically can anyone tell me what this is called so I can search for it. Of as said, just tell me it's all wrong to begin with.