The Iranian Revolution of 1979 was fought between Shah Pahlevi (supported by America and hated for his pro-American beliefs) and Ayatollah Khomeini (and anti-American Islamist). Khomeini succeeded in taking power from the Shah and created the Islamic Republic of Iran. During the turbulent years that followed, 300 militants seized 63 hostages from the American embassy in Iran and held them for 444 days.
Months of negotiations ensued and eventually, 11 of the hostages were released prematurely. The others were almost rescued by President Carter in Operation Eagle Claw, but due to a sandstorm, the operation was a disastrous failure. It was such a great failure that the US reformed its military and created a new unified combatant command, SOCOM, and a new special forces regiment, the 160th SOAR.
The Iranians remained steadfast in their imprisonment of the hostages until September 1980, when Iraq invaded Iran and launched the Iran-Iraq War. As a result, Iran wanted a quick resolution to the hostage issue and thus, the Algiers Accords were signed by the United States and Iran. The hostages were returned on Reagan's inauguration day. However, tensions between these two countries still exist today (see Iranian capture of 15 British sailors).
For more information, read: http://news.bbc.co.uk/2/hi/middle_east/3978523.stm
The American Presidents by David Whitney http://www.historyguy.com/iran-us_hostage_crisis.html Military History Podcast is sponsored by Armchair General Magazine