It's basically a cloture vote.
A three-fifths majority (60 votes) is required in the Senate to invoke a cloture in most cases. This's because the Senate reduced the number of votes required for cloture from two-thirds to three-fifths in 1975 since a two-thirds is very difficult to obtain.
However, invoking cloture on a measure or motion to amend the Senate’s standing rules would require a two-thirds majority (67 votes). Overriding a presidential veto still requires a two-third majority (67 votes), not only in the Senate but also in the House.
Also, the cloture vote majority does not apply to the House of Representatives - a supermajority would still require 290/435 votes.
You can read up more about the cloture vote in this CRS report.
This introduction on senate.gov explains this:
Three quarters of a century later, in 1917, senators adopted a rule (Rule 22), at the urging of President Woodrow Wilson, that allowed the Senate to end a debate with a two-thirds majority vote, a device known as "cloture." The new Senate rule was first put to the test in 1919, when the Senate invoked cloture to end a filibuster against the Treaty of Versailles. Even with the new cloture rule, filibusters remained an effective means to block legislation, since a two-thirds vote is difficult to obtain. Over the next five decades, the Senate occasionally tried to invoke cloture, but usually failed to gain the necessary two-thirds vote. Filibusters were particularly useful to Southern senators who sought to block civil rights legislation, including anti-lynching legislation, until cloture was invoked after a 60 day filibuster against the Civil Right Act of 1964. In 1975, the Senate reduced the number of votes required for cloture from two-thirds to three-fifths, or 60 of the current one hundred senators.
Also, this article titled "Why we need 60 votes for everything, in plain language" explains this in a clear way:
One of the basic rules of the Senate allows for unlimited discussion and debate. As long as someone wants to talk, discussion on a bill must continue. No vote can be held on a bill until all discussion/debate is closed.
If even one person wants to keep talking, debate on a bill continues.
The only way to make that person stop talking is when 60 Senators vote to end debate. This is called a cloture vote (cloture is from a word meaning to close off or bring to an end).
Cloture used to require a two thirds majority. This was back in the days when filibusters were rare. The cloture requirement was reduced from two-thirds to three-fifths in 1975--a more manageable 60 votes.
Regarding the repeal and replacement of the Affordable Care Act (ObamaCare) specifically, this article by The Washington Post describes it quite clearly.