You know that Git is a tool for versioning files. However, it has been built with collaboration in mind. In 2005, Linus Torvalds had the need for a light and efficient tool to handle tons of patches proposed to the Linux kernel from a multitude of contributors. He wanted a tool that would allow him and hundreds of other people to work on it without going crazy. The pragmatism that guided its development gave us a very robust layer to share data among computers, without the need of a central server.
So, a remote Git repository is nothing other than a remote copy of the same Git repository we created locally. If you have access to that host via common protocols such as SSH, HTTPS or the custom git:// protocol, you can keep your modification with it in sync.