Basically, Senate rules require 60 votes in order to stop debate (called a motion of cloture) on an issue, meaning that if a motion doesn't have 60 votes, the opposition can in theory continue debating the issue forever--or until the Senate is forced to go into recess. That's called a filibuster, and it has its precedents in Canadian politics too.
In practice, this doesn't happen all that often, but having 60 votes means that you can limit debate on any issue by forcing a vote on the bill at any time that you choose. However, a strict party-line vote in the Senate is the exception rather than the rule, so it's probably the case that more is made of the number 60 than needs to be.
As for "budget reconciliation," I don't know that one. It is true that differences between house and Senate bills are often reconciled in conference committee before it goes to the President... maybe this is something similar?