I think they mean that they'd like to be able to compare their scores with other peoples, friends, etc etc.
A lot of the time, people will just setup a server, then post the high scores onto that with HTTP requests. So you might have a friends list on the PC locally, then you can query the high score server and show the friends scores, top score, that sort of thing.
To be able to have a global, shared high score table, you need somewhere to keep it, either a PC of your own that is always on, that can be setup to act as a server - or a rented server, not hugely expensive as your high score file might not need to be very big, and database queries probably wouldn't be a big concern.
It's by no means straightforward, a bit of research is needed to find the best hosting option, but it's probably something that can be done. Probably best to leave it until the game is taking shape, no point worrying about it right now.

I am the one who knocks...
