Once you've got the GitHub integration set up and your user account connected to GitHub you can update issues from your commit messages. The commit message will be automatically added as a comment and if you want to update the status of an issue you can use any of the following commands followed by an issue number (i.e. "Close #2) to update an issue.
Resolve
Resolved
Resolves
Fix
Fixed
Fixes
Close
Closed
Closes
Note: Any form of "fix" resolves an issues, not closes it.