What is the correct way to make git ignore temporary files produced by vim in all directories (either globally across the system or locally for a single project)?

Vim temporary files end with ~ so you can add to the file .gitignore the line

*~

Vim also creates swap files that have the swp and swo extensions. to remove those use the lines:

*.swp
*.swo

This will ignore all the vim temporary files in a single project

If you want to do it globally, you can create a .gitignore file in your home (you can give it other name or location), and use the following command:

git config --global core.excludesfile ~/.gitignore

Then you just need to add the files you want to ignore to that file.

If you want to comment in a Git file you must do this on a separate line:

# Ignore vim files:
*~
*.swp
*.swo

Any comments placed on the same line as an active git ignore will cause the whole line to be misinterpreted.