Nonpartisan words and actions don't show allegiance to any political party. They're politically neutral.
In the United States’ political system, it often seems like politicians only do what's best for their party: the Democrats or Republicans. Politicians are partisan when they act in favor of their party. To behave in a nonpartisan way is to put politics aside. A nonpartisan bill wouldn't be good just for one party: it would be good for the whole country. Politicians who are nonpartisan stop worrying about elections for a minute and get some things done.