Appropriations - General Definition
An appropriation denotes an amount authorized in the budget for expenditure by an organization. Generally, an amount cannot be spent unless an appropriation is made (though some laws allow for spending in emergency circumstances, such as to recover from a natural disaster).
In state and local governments, the approved budget is also the appropriation authority. During the year, amendments are made to the budget, and thus appropriations, to account for updated conditions (such as increased revenue or to purchase an item not originally planned).
In the United States Government, though, the budget is not the appropriation authority; separate legislation must be passed to allow for appropriations. Absent the normal appropriation bills or emergency legislation allowing for continued spending (referred to as a Continuing Resolution), all "non-essential" work must cease (a "shutdown"). Congress has generally abrogated its duty to pass budgets and appropriations bills in recent years (specifically, the Senate; the House has generally passed the required legislation).