Short answers, pulled from the story.
HM Treasury is the United Kingdom's economic and finance ministry. It manages public spending, sets fiscal policy including the annual Budget, controls departmental spending limits through Spending Reviews, and coordinates economic strategy with the Bank of England. It also leads policy for the UK's financial services framework.
Despite being one of the smallest UK government departments by staff count, HM Treasury controls public spending across all other departments and sets the fiscal framework for the entire economy. Its power of the purse gives it authority to approve or constrain the budgets of every other ministry.
Budget comes from the French word bougette, meaning a small pouch or bag. A 1733 satirical pamphlet depicted Chancellor Sir Robert Walpole as a quack doctor opening his bag of tricks. The joke stuck, and the term shifted from meaning the container to meaning the financial plan itself.
The Stop of the Exchequer was a decision by King Charles II to halt all government debt repayments after heavy naval spending during the Anglo-Dutch wars caused government debt to balloon. The move substantially undermined trust in the government's finances and led Parliament to tighten its control over how public money was spent.
The Northcote-Trevelyan Report of 1854 reorganised Treasury recruitment around merit and fixed pay, replacing the old system of hiring through royal connections. Stricter entry examinations raised standards, though public school and Oxbridge graduates still dominated the Treasury's ranks by 1914.
HM Treasury is located at 1 Horse Guards Road in London. It moved there after Blitz bombing damaged the original Treasury building during the Second World War. The Treasury also has additional offices in Darlington and Norwich.