For people without write permissions to the GitHub oofem repository, the preferred way is to create a pull requests to propose and collaborate on changes in repository. The changes are proposed in a separate branch.
You can read more about Creating a pull requests on GitHub help pages.
Typically, one have to create a fork, or copy, of the repository. Forks let you make changes to a project without affecting the original repository. You can fetch updates from or submit changes to the original repository using pull requests.
If you have write access to the repository, you can also create branch directly within oofem repository. This should be reserved only to agreed, long-term, project-wide branches. For day-to-day and personal development, the forks are recommended.
After you have created a pull request, you can ask a specific person to review the changes you've proposed. In OOFEM, these people ('lieutenants') are in charge of a specific subsystem of the project and they merge in all changes related to that subsystem and push them to the reference (blessed) repository that everyone can clone from again.
Before contributing, please make sure you have followed oofem coding conventions.