A prime minister was a head of government, particularly of states ultimately ruled by a king or queen. [source needed] The title derived [source needed] from the fact that they were the leading minister of a larger group of ministers — called, in Britain, at least, a "cabinet". [source needed]
As the leader of that group, they had chief fiduciary responsibility to the sovereign, and therefore enjoyed frequent communication with their monarch — as could be seen in the relationship of, say, Richelieu with Louis XIII. (AUDIO: The Church and the Crown) Additionally, the prime minister held the responsibility of setting the Cabinet's agenda and speaking to the public on its behalf. (TV: World War Three, Victory of the Daleks)
Earth
United Kingdom
On Earth, the United Kingdom was headed by a Prime Minister. The Doctor had numerous encounters with various British Prime Ministers, most notably Winston Churchill, who encountered several of his incarnations, (TV: Victory of the Daleks) Harriet Jones, who was deposed by him, (TV: The Christmas Invasion) and Harold Saxon, which was an alias used by the Master. (TV: The Sound of Drums/Last of the Time Lords) During a frozen planes crisis UNIT called Coal Hill School in an attempt to reach Clara Oswald, however Mr. Dunlop picked it up and they almost put him through to the prime minister.
India
The Indian government was also headed by a Prime Minister. In 1984, the position was held by Indira Gandhi. As she told the Egyptian Pharaoh Erimem, Peri Brown considered her to be an effective female leader in contrast to her British counterpart and contemporary Margaret Thatcher. Given that Erimem was from the 14th century BC, she had unsurprisingly never heard of either Gandhi or Thatcher. (AUDIO: The Eye of the Scorpion)
In the wake of Miracle Day in 2011, the Prime Minister of India called for peace with Pakistan. (TV: Rendition)
Extraterrestrial occurrences
During the New Dalek Paradigm, the Daleks were led by the Parliament of the Daleks, with the head of Parliament being the Prime Minister of the Daleks. (TV: Asylum of the Daleks)
Delta IV, a planet in the United Federation of Planets, had a government headed by a Prime Minister in 2368. (COMIC: Assimilation²)