The 60 votes rule is usually referenced for a procedure called "cloture." The policy of the Senate is unlimited debate before there is a vote on a bill. However, if 3/5 of the Senators vote to end the debate (cloture), the debate ends within a specified number of hours, and the bill is brought to a vote.
You're right, it's hard to call cloture. But it's supposed to be hard. Because the Senate has historically been designed to be slower and more deliberate than the House of Representatives.
The reason why you may have heard that the Senate needs 60 votes to pass anything is because, without 60 votes, the minority in the Senate could "filibuster," which means they could just endlessly debate and never allow the issue to come to a vote. This usually happens on controversial issues (like tax increases or spending on the War in Iraq).