Non-essential
Most government departments are still expected to perform certain tasks during a shutdown. They have been forced to divide their workforce into those essential, and those who are non-essential. Non-essential workers have been placed on furlough, while essential workers continue to report to work. The official government language has been changed to "excepted" and "non-excepted" employees, but people in Washington and in the media continue to refer to essential and non-essential workers.
Continuing resolution (CR)
In recent years, Congress has been increasingly unable or unwilling to pass a full annual budget. Instead they have passed continuing resolutions, which fund the Government for a shorter period of time. The prevalence of short-term CRs has led to more frequent, and often highly contentious, debates over government funding. The Democrats are currently pushing for a "clean" CR, meaning a law designed simply to fund the Government with no additional measures attached. Republicans are attempting to tie the CR to funding for Obamacare, a tactic which the Democrats have repeatedly rejected. Both sides anticipate a CR which will fund the Government for as little as six weeks, indicating that the next budget showdown is not far off.
Debt ceiling
The debt ceiling is the cap on the amount of money that the Government can borrow. It first came about in 1917, and since 1960 the Government has raised it 78 times. The debt ceiling is currently set at US$16.7 trillion ($20 trillion), but it is estimated that this limit will be exceeded on October 17. Should the debt ceiling not be raised by that time, the US will begin to default on its debt for the first time in its history. When that nearly happened in 2011, it caused the US credit rating to be downgraded. Small-government Republicans have indicated that they will demand concessions from Democrats on spending in order to raise the debt ceiling, while President Barack Obama has said he will not negotiate on this issue. If the debt ceiling is not raised on time there could be catastrophic economic consequences.