Term used in the Parliament, the government of the UK.
Before we embark on a lengthy explanation on what a three-line whip is, let's begin with examining what a Whip is.
A Whip is akin to a middle manager for all MP's from a party. In military terms, he or she could perhaps be called a sergeant. This person is crucial to the execution of the "parliamentary business", which is the rules governing the operation of the Parliament.
One of the responsibilities of the Whip is to make sure the MP's vote like the party wants in important divisions. Division is the term used to describe the way the House of Commons actually cast their votes. Those voting "No" line up and parade through the lobby to the left of the Speaker, while those voting "Yes" (or "Aye") walk through the division lobby on the right.
It might dawn on you now why those persons hanging around the government all day are called lobbyists. But I digress.
One of the more important responsibilities given to a Whip is to make sure the MP's all vote in line with the party on particularly important divisions. In order to prepare the MP's for the upcoming week of parliamentary business, the Whip writes an agenda (or "Whip"). This piece of paper distributed to all MP's lists the divisions of the upcoming week, each underlined once, twice or three times to indicate the importance of each.
A division underlined three times (that's the three-line whip right there!) means that the party expects the MP to be present to vote. Failing to show up for the three-line whip is considered very serious matter. A party can withdraw the Whip from the MP, leaving him or her isolated in parliament. This more or less leaves the MP as independent.