To clone a GitHub hosted repository to local computer:
# git clone $RepoURL.
To get the RepoURL you need to go to the repository page on GitHub and copy the URL (HTTPS: https://github.com/…/$ProjectName.git, SSH: firstname.lastname@example.org:…/$ProjectName.git). In this way you will get the repository copied to a sub-directory of the current working directory. The sub-directory has the same name of the repository.
You don’t need to initialize git in the folder because it is already initialized (.git folder contains the git information).
The remote is a reference to repo. To list the references, execute: # git remote -v. You will see the automatically created references (origin). To add a reference to the origin: # git remote add origin $RepoURL