I think we need more context. I am familiar with the term in relation to US politics, but you were reading a book on economics. In addition, the procedures are different in different countries, so we should know what country was being discussed as well.
Could you please give us the sentence in which it appears, and perhaps a sentence or two above it? This will help us to be sure that we are answering accurately.
Mr Obama chose a Republican, Judd Gregg, as commerce secretary.
Mr Gregg is a senator from New Hampshire and agreed to take the job only after assurances that the state’s Democratic governor would pick a
Republican to replace him.
The Democrats are tantalisingly close to a 60-seat filibuster-proof majority.
A filibuster is a speech or a person making a speech in a legislative assembly which is not designed to instruct anyone, or present a point of view, but to hold up the business of the assembly. Someone might do it to prevent the assembly discussing other matters which the filibuster does not wish discussed. Often assemblies have rules that if a matter is not discussed within a time-frame, it has to be thrown out.
This has nothing to do with Economics at all.
The sentence suggests that if you have a 60-seat majority you have a means of stopping people from filibustering.
When a minority in the Senate opposes the passage of a bill that the majority will pass if it comes to a vote, they sometimes filibuster. The Senate rules say that anyone can speak as long as she or he wishes, and as long as a Senator is speaking, they cannot call for a vote. Sometimes these opposition Senators just take turns speaking as long as they can to delay the vote: this is called a filibuster. It doesn't matter what they are talking about, they can just keep talking.
The only way the Senators can be stopped from speaking and forced to allow the bill to be voted on, is for at least 60 of the 100 Senators to vote for "closure". If President Obama has at least 60 Senators on his side, they will always be able to end the filibuster: he has a "filibuster proof majority." If he had 51-59 Senators on his side, he would still have a majority, but it would not be filibuster proof.
Thomas Tompion has written a more succinct explanation as I was writing this, but I will post this anyway.