Yes. You can place a git repository anywhere - including the invisible .git directory created by another git repository. I have a friend who has git track all his system config files in case he makes a mistake.
When working on a project, you want to init your repository in the root directory of the project.
To elaborate, each "working copy" of a Git repository is itself a Git repository. If you have a remote copy on a server, that is also a repository. You don't "check out" from there - rather, you "push" your changes and they are merged. If working on a purely personal project, the remote repository is often unnecessary. If you do want to host remotely, Github is a good, free, public choice.