Rufus Scrimgeour is the Minister of Magic that follows the disgraced Cornelius Fudge in the Harry Potter series of books. His previous job was head of the office of Aurors, the special police force that tracks down Dark Wizards. His ascendancy to the ministership is due to the fact that the wizarding world is now at open war with Lord Voldemort, a situation for which a former Auror would be well suited.
The sixth book of the series, Harry Potter and the Half-Blood Prince, opens with a conversation between Minister Scrimgeour, and the muggle Prime Minister of Great Britain . In the scene, Rufus is shown as a much more energetic, perhaps even severe, person. However, we soon learn that he has also had disagreements with Albus Dumbledore, which is usually a sign of foolishness. Later on in the book, Rufus personally meets Harry Potter and asks him if he would make appearences to help people's belief in the Ministry of Magic. Harry refuses, thinking that Scrimgeour is more interested in appearing to win the war than in actually doing so. Scrimgeour's appears again at the end of the book, but his ministership is does not appear again as a key element of the plot.
When I began to read The Half Blood Prince, I was surprised but very interested with the first chapter, featuring the meeting between Scrimgeour and the British Prime Minister. In the previous two books, JK Rowling had taken us beyond the world of Harry and his friends to show us more of the larger issues in the wizarding world. Before this book was released, and while reading the first two chapters, I imagined this book would continue in a similiar vein. Soon, however, the book narrowed down to showing the basic conflict between the central characters. However, Rufus Scrimgeour and his conduct of the war are still mentioned several times.
It is possible that they were originally intended to play a larger role, or that they are foreshadowings of events in Book VII. Another possibility, that would involve quite a bit of speculation, is that JK Rowling inserted this character as a comment on the conduct of historical wars, past and present. Albus Dumbledore and the rest of the Order of the Phoenix are made heroes in the book, after all, for aggressively pursuing a course of war against public denials of a threat. Most of JK Rowling's readers, and probably Miss Rowling herself, however, are probably not very large supporters of war or militarism in general. It could be that when Harry Potter speaks out against the random arrests and imprisonment of innocent wizards on trumped up charges, so that the government can appear to be doing something, JK Rowling is making a subtle statement about current events.