A Senate rule required a super-majority of three fifths to move to a vote through a cloture motion, which closed debate on a bill or nomination, thus ending a filibuster by a minority of members.
http://en.wikipedia.org/wiki/Supermajority
The cloture rule originally required a super-majority of two-thirds of all senators "present and voting" to be considered filibuster-proof.
For example, if all 100 Senators voted on a cloture motion, 67 of those votes would have to have been for cloture for it to pass; however if some Senators were absent and only 80 Senators voted on a cloture motion, only 54 would have had to vote in favor.
However, it proved very difficult to achieve this; the Senate tried eleven times between 1927 and 1962 to invoke cloture but failed each time.
Filibuster was particularly heavily used by Democrat Senators to block civil rights legislation.
In 1975, the Democrat Senate majority, having achieved a net gain of four seats in the 1974 Senate elections to a strength of 61 (with an additional Independent caucusing with them for a total of 62), reduced the necessary super-majority to three-fifths (60 out of 100).
However, as a compromise to those who were against the revision, the new rule also changed the requirement for determining the number of votes needed for a cloture motion's passage from those Senators "present and voting" to those Senators "duly chosen and sworn".
Thus, 60 votes for cloture was necessary regardless of whether every Senator voted.
The only time a lesser number would become acceptable is when a Senate seat is vacant. (For example, if there were two vacancies in the Senate, thereby making 98 Senators "duly chosen and sworn", it would only take 59 votes for a cloture motion to pass.)
The new version of the cloture rule, which had remained in place since 1975, made it considerably easier for the Senate majority to invoke cloture.
This considerably strengthened the power of the majority, and allowed it to pass many bills that would otherwise have been filibustered.
The Democrat Party held a two-thirds majority in the 89th Congress of 1965, but regional divisions among Democrats meant that many filibusters were invoked by Democrats against civil rights bills supported by the Northern wing of the party).
Some senators wanted to reduce it to a simple majority (51 out of 100) but this was rejected, as it would greatly diminish the ability of the minority to check the majority.
The three-fifths version of the cloture rule does not apply to motions to end filibusters relating to Senate Rule changes.
http://en.wikipedia.org/wiki/Cloture#United_States
