Working with Git locally

Wasn't sure where this thread should go so added it here.

In VS, you can choose TFS or Git as a version control when you create a new project.

When using Git, can this be done locally, so all your different versions of files are pushed to a repository on your local machine or do you have to use github.com's server and push there?

If you want to use TFS, is there a free version or do you require a license to use it?

Thanks.