The basic story goes like this. On November 4, 1979, Iranian radicals stormed the American embassy in Tehran, taking 52 Americans hostage. The hostages were held for 444 days, and not released until minutes after Ronald Reagan was inaugurated as president.
However, as always, there is more to the story.
The storming of the embassy came about eight months after the Ayatollah Ruhollah Khomeni seized power in Iran after the outser of the country's leader, Shah Mohammed Reza Pahlavi. But the Shah had taken power after a coup d'etat by royalist officers backed by the United States and Great Britain deposed prime minister Mohammad Mosaddegh in 1953, resolving an ongoing power struggle between the two.
While the Shah's rule saw the Iranian economy improve significantly, he soon backed a secret police known as SAVAK. The growing repression, though, helped make the Shah more and more unpopular. By January 1979 he and his family left on a "vacation" and never returned.
After the Americans were taken hostage, Khomeni gave his approval to the capture of the U.S. embassy in Tehran. Following that, the U.S. attempted a hostage rescue mission that failed. The crisis crippled then-President Jimmy Carter, who eventually lost to Ronald Reagan in a landslide.
Reagan's administration would proceed to launch a major peace-time military buildup, and the eventual end of the Cold War.
This History Channel video discusses the hostage crisis and its impact on U.S. policy in the decade that saw the end of the Soviet Union.