As for any other country, in US as well all the goverment expenditures must be authorized by the House, which is Congress (Congress consists of the House of Representatives and the Senate). If these provisions for funding the government operations are not authorized and provided for the government will simply not have sufficient funds to run. And eventually stop operations. This is what happened this Tuesday October 1st, 2013. The Government shutdown in the US of 2013 is such situation where congress was unable to come to a consensus over discussed bills and had to shutdown.
US Congress consists of 2 major political parties which are Democratic Party & Republican Party. Obama is a democratic president. This shutdown is caused due to both parties could not hold similar opinions over spending bill to fund the government. The two parties argued for over a week on Obamacare, the president's signature health care law. Republicans wanted that spending bill include anti-Obamacare amendments.
All the essential services including weather agencies, military, medical, air traffic etc. will continue to run except non essential ones.