In the Senate (both in the US Senate and here in Missouri), the senators debate a bill until there’s an agreement that the debate’s over. Only then a final vote can be taken. The vote to end debate (called a cloture vote) requires a supermajority, while a final vote generally requires only a simple majority to pass. A filibuster is an attempt to prevent a bill from passing by keeping it from ever coming to a vote.
Jimmy Stewart’s character in “Mr. Smith Goes to Washington” made the filibuster famous by singlehandedly stalling the entire Senate. Fortunately (or unfortunately, depending on how you view the bills being passed) the movie is misleading; a single Senator can’t hold up a bill that way. S/he needs 39 accomplices in the US Senate to go along – otherwise, the rest of the Senate would vote for cloture, bringing the debate to an end.
Nowadays, a full-fledged filibuster hardly ever happens, with all the images it conjures up of talking to all hours of the night reading from the Federalist Papers, recipe books, the Bible, and letters from home. The filibustering side threatens, and if they have the votes, the bill is usually withdrawn without so much as a cloture vote.