When you use a version control system to keep track of your changes to a set of files, there are two different concepts you're dealing with: a working copy and a repository.
A working copy is just a folder with your project's files in it. It contains the versions of the files that you are working with now.
A repository is a record of the entire history of your project. When change something in the working copy, you can then commit it to the repository and your change gets added to the history.
You can also go back and grab an older version from the repository if you find you need it. (If you do this, then the working copy will be updated so as to contain that older version rather than the most recent one.)
Older centralised version control systems use a separate database for the repository. But with a distributed version control system such as Mercurial, the repository – the entire history of your project files – is stowed into a special folder inside the working copy on your hard drive. Every change you commit gets added to the history in that hidden folder. (The history is compressed, so it doesn't take as much space as you might imagine.)
The term remote repository simply refers to a repository related to your local one, but stored on another computer somewhere else. Often this may be a “master copy” of your project stored on a server elsewhere, which you and your collaborators can use to keep up with each other's work, or which you can use to make your work public, or simply use as a private backup.