The word "filibuster" comes from the Spanish word "filibustero," meaning "pirate," or "buccaneer." Filibusters were authorized by a Senate Rule in the early 19th Century by Angry Conservatives who were unhappy that they couldn't silence the voice of the people easily enough.
Filibusters were particularly useful to Southern senators who sought to block civil rights legislation, including anti-lynching legislation, until cloture was invoked after a fifty-seven day filibuster against the Civil Rights Act of 1964. In 1975, the Senate reduced the number of votes required for cloture from two-thirds to three-fifths, or sixty of the current one hundred senators. 
Filibusters are when a Senator stands up and begins to talk, and never stops. This delays bringing a bill to a floor vote that would obviously pass. They can be stopped only by a closure vote, which needs sixty yay's. Filibusters can be used constructively, if used sparingly, such as Mike Gravel's miraculous five-month filibuster to help end the draft during the Vietnam War.
In recent history, Senators stopped using talking filibusters, and just declared they wanted to fuilibuster a bill, and the bill was filibustered. Under Mitch McConnell, nearly every bill has been filibustered, so instead of needing 51 votes to pass a bill, you need 60, even though the U.S. Constitution calls for majority rule in the Senate except on certain issues, for example, impeachment.