When I make changes to my local branch, and commit them, what do others have to do to see those changes in their local repositories? What does 'git checkout' do?
First of all, I recommend you to check the official Git's Getting Started Guide in order to learn further how Git works.
Answering your question, you first need to understand that all changes you make on your local computer won't be seen to others unless you push your code to a remote repository (where others could pull these changes to their local machines from).
The git checkout command will let you switch from branches.
There's more info and practical examples on the git docs.