A political football is a political topic or issue that is continually debated but left unresolved. The term is used often during a political election campaign to highlight issues that have not been completely addressed, such as the natural environment and abortion.
There are many reasons that an issue may be left unresolved. Examples are:
With the use of the word football, many other football terms are in common use. Once a topic or issue has presented itself, the opposition will take the issue and "run with it" and "try to score points". There may be many parties in opposition to take up the topic, and sometimes run it into the ground, which at times can "backfire" and end up hurting the opposing faction. This can happen to the point of that party being seen as "fumbling" the issue. When a party or candidate ends up ignoring or putting off the issue, they are said to be "punting" it. In addition to the opponents of the politician or party using the subject as political fuel, the media may also "run with it". Sometimes these matters become larger, like during an election, than anyone would have thought possible during the event itself.
The phrase in modern usage is most likely derived from American football rather than association football ("football" to most of the world). However, the Oxford English Dictionary has examples of use as early as the 1600s (long before the creation of American Football). All early examples from the OED originated in the United States. It is possible that the term originated from one definition of football and evolved later to be strongly tied to the modern definition of American football.
The term is now being used around the world including even places where most people do not know much about American football or the relation to the term.