Being from England, I had no idea what the hell this was and so I'm answering assuming you knew as little as me. So this summary is for people from the non-US who didn't know this was possible.

A budget plan has to be drawn up before the start of the Fiscal year, which was October 1st, otherwise we don't know how much is to be allocated to each sector. Congress debates how much money goes into each sector before the start of the year.

This is normally fine, but because you've got Republicans in the House of Representatives and a Democrat in the White House, you get a lot more fighting.

This year they missed the deadline because they were arguing about Obamacare. The lack of resolution over this bill means throwing the entire baby out with the bathwater and no one gets paid because the budget wasn't allocated. As your Government assumed this would happen from time to time, they set up some rules that dictates who gets paid and who gets told to go home whilst the country sorts out who gets paid what. Taking from bfv's document, these are the people exempt and still getting paid. Everyone else gets 'furloughed' : legal temporary unpaid leave.

1.Employees who are necessary to address emergency situations where the failure to perform those functions would result in an imminent threat to the safety of human life or the protection of property.

2. Employees who perform functions that are funded through fees or under multi-year (as opposed to annual) appropriations.

3. Employees who perform functions that are related to express authorizations to contract or borrow without an appropriation.

4. Employees necessary to meet the obligations necessary to the discharge of the President’s constitutional duties and powers. This is understood to be employees necessary to interpret statutes, such as the Antideficiency Act, to avoid significant constitutional issues.

5. Employees who are required for the orderly termination of agency functions.

So non-essentials are trimmed whilst Congress sorts it out. Longest was 21 days in 1995, The Independent works out about $10BN in cost per week to the US economy.