"The British Ministry" Patrick Henry talks about in his speech is the ministry (or administration) headed by Lord North, who was Prime Minister of Great Britain from 1770 to 1782…which is the exact time in history it was probably worst to be Prime minister of Great Britain.
Born into a wealthy and titled family, Lord North had a generally successful early career in politics. While people who didn't know him well thought he seemed pretty boring, he was actually a witty and talented speaker, which served him well in his political career. While he started out as a Whig (or liberal), he slowly slid to the right over the course of his life, ending up as a Tory (or conservative) by the time he took office as prime minister at the age of thirty-eight.
Like many politicians before and since, North inherited problems that existed before he took office and got blamed by history for those problems. Things were already looking rough by 1770, but then North made it worse by passing the Tea Act, which led to the Boston Tea Party. (Check out the "Timeline" for more.)
North wanted to enforce the taxes that had been passed and leaned toward using the stick rather than the carrot, but he went against his own instincts and tried to make friends with the colonies in 1775.
It didn't work.
North's job during the war was mainly to deal with funding rather than the actual military operations, and this was a guy who was such a fiscal conservative that he made "Budget Day" a big deal in Great Britain's political calendar, so we imagine he wasn't thrilled to pay for it.
Things started going south for North when the British suffered major defeats at Saratoga and Yorktown. The House of Commons returned a vote of no confidence in North's ministry in 1782, and he resigned with his cabinet. (Cue the sad trombone noise.)
This was the first time a prime minister had resigned after a vote of no confidence from the House of Commons and indicated changes in British government as well, in that the common people were taking a lot more control than they'd ever had before.
While North is mainly remembered today as the man who lost the American colonies, that view ignores the fact that he did maintain a stable government through a very turbulent period.