Once you've created your account you can check your user name by selecting the icon in the upper right corner and then selecting "Your Profile" (select the link at the bottom of the screen to skip the tutorial first). Your user name should be on the left side of the screen.
Contact the administration of the Github repository you need access to and include this username in the message (in this example the username would be "gresh-test"). The administration can add you to the project and you should receive a confirmation email. Follow the instructions in the email.
"Cloning" the repository just means pulling a copy of the files down to your local machine in a way that any changes you make can be merged back into the repository.
After you've accepted the invitation you will be sent to the home page of the repository. Copy the url.
You'll want to follow the repository owners instructions for forking and branching, but for now, lets just look at the case of working directly on the main branch of the existing (unforked) repository (which you should never do... ever).
Open a cmd prompt and navigate to the directory where you would like the project to live.
After you've install the client you will need to issue the following two commands (this is usually a one time thing).
git config --global user.email "the-email-you-registered-withc@yourhost.com"
git config --global user.name "YOUR_USERNAME"
To clone the repository, enter the command:
git clone <paste-repository-name-here>