[Fegasderty]

I've read the full "Operation Ajax" post-op assessment referred to in abrahamsen's link. Here are the highlights.
Iran tried to nationalize oil fields which were developed by BP, then known as the Anglo-Iranian Oil Company. The idea of reaching a negotiated settlement was briefly entertained, but British figured did not want the notion of nationalization spreading; the idea was to enforce the terms of its contract.
At the time, Iran was politically structured something like the UK - there was a monarchy and a parliament (called the majlis). The monarchy was losing influence to the parliamentary system in place, but the Shah Mohammad Reza Pahlavi and his family weren't exactly fighting to maintain supremacy; times were changing, and the Shah was allegedly plagued with self-doubt (he fled Iran repeatedly over the years when he perceived a threat).
The US, however, judged that if it were possible to force a public conflict between the Prime Minister (Mohammad Mossadegh) and the Shah, then the military and public would side with the monarchy. The strategy to overthrow Iran's parliamentary system therefore war born - the goal was to force a conflict and let the chips fall into place.
Long story short, the Shah didn't want to participate initially; he in fact, ran away in order to avoid signing a decree to dismiss the Prime Minister. There is no smoking gun proof, but I read parts of the clandestine service manual insinuating that the Shah was threatened by US officials; that one way or another, the US and UK would move forward. Don't take that as fact though. Suffice it to say, a lot of arm twisting was had before the Shah finally conceded. In fact, I think at one point the US was forging letters by the Shah.
The attempt almost failed, and as the Americans tell it, the CIA people were preparing to evacuate the US Embassy. Things then got very messy in the media, and it seems the US was unable to totally understand what was happening. The big mistake allegedly came when Mossadegh loyals came out condemning the Shah and making threats, following the initial coup attempt (there were tanks and everything); this is what polarized the military.
Mossadegh was overthrown and the Shah was installed. However, as I understand it, any undemocratic violation of a country's political structure necessarily requires brutality to enforce the new status quo; the Shah's secret police (trained by the US) captured and tortured political opponents. This was well known in Iran it seems but even the Shah claimed to be unaware of such things in some television interviews; I almost believe him.
That was 1953. In 1979, the people revolted. It was initially a socialist democratic islamist revolution, but it was the ayatollah Khomeini who led the revolution from exile, as its figure head. Once the Shah was overthrown, the islamists betrayed their revolutionary partners and agreements. Civil society was gutted; religious persons, regardless of qualification, were inserted into authoritative posts (heads of hospitals, judges, etc). You can read about this era in Shirin Ebadi's autobiography. This was the birth of the today's Islamic republic of Iran. Shortly thereafter, the Iran-Iraq war began, and galvanized the public behind its new leaders.
The revolutionary government's vision for spreading Islamic (Shi'ia) revolution around the middle east is also important for understanding the Arab spring, the civil war in Iraq, what's going on in Syria. Sectarianism is on the rise, and Arab nations (who are traditionally closer to the US) view a powerful Iran as a threat to the established power structures, which revolve around Sunni supremacy. According to Nir Rosen, Sunni and Shi'ia managed to share power reasonably well under Saddam Hussein; the US in its infinite wisdom though decided to re-create a new political class, handing over the reins of governance to the Shi'ia minority and dispossessing the Sunnis. This caused a civil war, which when read in the minutae of Rosen's field notes, overshadows the US presence in Iraq so much that it almost seems like an afterthought.
The dynamics are very complicated.