A remote repository is a copy of a project kept somewhere your computer can connect to, often on a server. It lets people share changes and keep a copy away from their own device.
Think of your computer's project as your own notebook and the remote repository as a shared copy. You can send your changes there or bring other people's changes back.
The remote may be hosted by a service such as GitHub or GitLab, but the remote repository is the shared project itself, not the hosting company. Sending changes is a separate action from saving them locally.