A prime minister is the elected leader of a government. In 1960, the Sri Lankan politician Sirimavo Bandaranaike was the first woman to be elected prime minister of any country.
One major difference between presidents and prime ministers is that a president is officially "head of state," while a prime minister is "head of government." In monarchies like the United Kingdom, the prime minister serves the Queen, the actual head of state (although that's become a ceremonial position over time). Countries that were once monarchies, like Germany, have both a president and a prime minister, though the latter holds the real power.