(covers information from several alternate timelines)
A budget was an amount of money set aside for a specific purpose.
In an alternate timeline February 23, 1936, the mayor of New York City approved a budget increase. The story was reported on by The Star Dispatch, who noted that City Council had begun a debate on changes to the budget. (TOS: "The City on the Edge of Forever")
In 2024, District Security had their budget cut for the umpteenth time. Lee warned a time travelling Benjamin Sisko of this when he sought help from her in a Sanctuary District in San Francisco. (DS9: "Past Tense, Part I")