To take an issue that is non-partisan and turn it into a partisan one. Political parties do this to gain an advantage over the other party, tossing an issue back-and-forth as a “football.”
The term can refer to the game of political football, where parties do this, or the issue itself. In the former case, parties are playing political football; in the latter, the issue is a political football.
The first known use of the term dates back to 1857, where it was used in a Maine newspaper.