A remote repository is generally a copy of a local repository that exists on a server somewhere that other people also have access to.
There are two points to having a remote repository:
It acts as a backup for the local repository
It allows a team of people to collaborate on a Git project
To state the bleeding obvious—local repositories are, well, local; they live on a specific computer used by a single project team member. The local repository is for the use of that specific person on that specific machine.
A remote repository on the other hand is hosted on a server somewhere and can be accessed by all the team members.
Local repositories have associated working and staging areas where work is developed staged and ultimately committed to the local repository.
Remote repositories do not have working and staging areas; it is just the bare repository.