Refers to a person who previously held the office of prime minister. Often used in news or formal contexts.
The former prime minister held a press conference.
Also means 'former prime minister', but 先の総理大臣 is slightly more formal and often used in news reporting, while 元総理大臣 is more common in everyday conversation.
Compound of 先の (saki no, 'previous') + 総理大臣 (sōri daijin, 'prime minister').