When you encounter an error while pushing to a remote repository because it contains changes you don't have locally, you need to integrate those changes first. Start by fetching the latest updates from the remote and then merge or rebase them into your local branch. If conflicts arise, resolve them manually in your text editor, stage the resolved files, and complete the merge or rebase process. Once your local branch is up-to-date and conflicts are resolved, you can safely push your changes to the remote repository.