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Other people have produced tutorials on Version Control, Basics of using Git, and workflow using Git.
Differences between GitHub.com and The University of Texas’s repository (github.austin.utexas.edu)
GitHub.com is a location where you may store your open repository for free and you can explore the open repositories of other individuals and research teams that archive their materials there. For example, I just found a repository for DHS indicators, which provides stata "code for all DHS Program indicators listed in the Guide to DHS Statistics" by searching for "demography." It is possible to pay to create a private repository should you not want to make your code openly available immediately. Note that even open repositories are only readable to the world. One can't alter an open repository without becoming a collaborator, but one can "fork" a repository to create a copy for oneself that remembers its origins.
The University of Texas system repository is designed to be used only by UT employees and is a secure location for code that you want to keep within a team of people employed by UT-Austin (including people with zero-time appointments). Each person needs to login to github by going to github.austin.utexas.edu and using their uteid and password to login. After that one can create and be added to existing github.austin.utexas.edu repositories. Note that people not employed at UT cannot access the files in this repository. You can (and should) move a repository to GitHub.com when you leave UT Austin.