Appropriations: The Constitution says "no money shall be drawn from the Treasury, except in Consequence of appropriations made by law." That means Congress must write and pass a bill to spend every dollar that goes out of the federal Treasury.
Budget: Appropriations are not the same thing as the budget. The budget is a plan for spending for the next year. Appropriations actually spend the money.
Continuing Resolution: These are short term spending bills that Congress can pass to keep the government funded for a short period of time.
Do Republicans have the votes they need to pass the appropriations bill on their own?
A: Not exactly. The bill is subject to the cloture rule in the Senate-- so it requires 60 votes to close debate, and so 9 or 10 Democrats are needed, depending on how many Rs are in DC
Short history of the Appropriations Committees in Congress, that have been writing funding bills since 1865: democrats-appropriations.house.gov/sites/democrat… :)