Received and logged
The invoice is recorded with the date it arrived. If something is missing, the government may return it or ask for a correction. Ask what is missing and resubmit promptly.
Approved by the department
Someone in the department that ordered the work confirms that it was received and that the price matches the order.
Reviewed by the finance office
The finance office checks the account to be charged and that money is available, then prepares the invoice for payment or for the board.
Allowed by the board, where required
Many townships, cities and counties pay claims only after their board approves a list of them at a public meeting. An invoice that misses one meeting's list waits for the next.
Paid
Payments go out in batches, by check or bank transfer. Where your state's prompt-payment law sets a pay-by date, it counts from the dates above.
Four questions to ask
Which step is the invoice at? Is anything missing? Which meeting's list is it on? When is the next payment run?
This is general information, not legal advice. Steps and names differ from one government to the next.