A filibuster is a common tactic used in the Senate, where debate time is unlimited, to prevent the legislation on the floor from passing. It is part of the Parliamentary Procedure in that chamber of Congress. It requires 60 senators to break a filibuster in what is known as cloture. Filibusters are impossible in the House because the House has adopted rules to permit a majority of the members to end debate on a matter in response to the pressures created by the size of the body. The House of Representatives was able to do this, despite opposition from minority parties, because it is not a continuous body, it must reorganize itself and adopt rules from scratch by majority vote after each election. In contrast, the Senate is a continuous body which only amends its rules from session to session, which requires a two-thirds majority.
The "nuclear option", which has been considered by Republicans in the U.S. Senate would be to have the parliamentarian declare that the filibuster requirement is unconstitutional in light of an implicity constitutional understanding that the Senate operates by majority rule. This ruling would stand if affirmed by a majority of Senators, under Senate rules, unless the Courts chose to intervene on this break with long time practice, which would be itself unprecedented. Majority parties have been loathe to consider such a "nuclear option", however, because it would fundamentally weaken minority parties going forward, and Republicans recognize that they too may someday become a minority party. One reason that minority parties use the filibuster sparingly, rather than as a way to veto every close issue upon which the majority seeks to act (e.g. Democrats have confirmed roughly 98% of George W. Bush's judicial nominations, despite the fact that they could have filibustered them and despite the fact that Democrats often dislike the administration's choices, leaving the filibuster for the most extreme cases) is to temper the majority's desire to invoke the "nuclear option."
Prior to the 20th Century, most filibusters were used by Southern Democrats to oppose civil rights legislation. The ability of the Senate to hold any kind of cloture vote is itself, historically recent.