If you've been invited to be a committer on the project, congratulations! There are a number of steps you must follow to be able to merge patches to the project.
For most pull requests (PRs), we use a merge tool which squashes multi-commit PRs, creates a nice commit message (including the PR description, which is valuable for the changelog), and updates the corresponding JIRA status. Read https://github.com/apache/arrow/blob/master/dev/README.md for instructions how to set this up.
If you run into any problems pushing to the Git repository, it's possible that you have not been added to the project committer roster by the PMC chair. If this happens, please ask the PMC chair to double check that you have committer karma.