One of the dumber things I've read about Iran in the last few days was a fellow asking why the Iranians can't "get over something that happened in 1953". It was stupid because he was complaining about the protesters, whose hopes for more normalized relations with the United States are certainly a step towards "getting over 1953". Also, the comment revealed the mentality of those living in the eternal present- failure to forget the past is seen as some sort of character flaw.
Anyway, the best way to explain why people still care about the 1953 coup is this: The American people elected Dwight D. Eisenhower in 1953. However, some of the economic steps he took offended... say, Iran and Britain, who wanted to have stronger control over American industry. So, they took a guiding role in orchestrating a coup d'état to remove Eisenhower and replace him with a hard line ruler*, whose secret police then spent over a quarter of a century repressing all forms of dissent through torture, intimidation, and even murder, until finally, there was a violent revolution to remove him, followed by more purges, violent intimidation, etc. etc.
It's critical to remember all of this because even before the coup, the protests against Mosaddeqin were orchestrated by Kermit Roosevelt's boys- they simply paid the protesters to march against the Prime Minister and stirred up the clerics against him. None of this is a secret either. At the time, it was seen as a great success- the model for future CIA involvement in the world. More importantly, everyone in Iran knows about this, which is why the protesters now- who I fear are about to be repressed yet again- are so adamant about not being associated with the US. The regime will call this a foreign-backed coup, and the situation will, yet again, get a lot bloodier before it becomes "peaceful".
Again, I'm thinking of Prague Spring watching this and afraid of what comes next. I'm also thinking of Tacitus's classic description of tyranny: ubi solitudinem faciunt, pacem appellant. "where they made a desert, they called it peace."
*I know that doesn't quite work with the Shah, but I can't think of an American equivalent.