Pushing to a remote : send some commits you have to a another git repo. The git repo is considered as “remote”, but it can be a repo in another folder of your hard drive.
pulling from a remote : get some commits from a remote repo and merge them in your current HEAD (your current checkout of your repo)
Your coworker might have use pull instead of push because your repository might not have been available (no git daemon running, or gitweb, or ssh server on), but his was avalaible from your computer. As it is a server, he might not want to expose a git daemon/service which could be a vector of attack.
But if your repository was shared/available, he would just have been able to do :
- change something locally
- commit
- push to your repository