True so far, Reza Shah had ties to Nazi Germany, leading to fears he might join them. Whether that was true or not is another discussion and is outside the scope of this topic.
Prime Minister Mohammad Mosaddegh tried to usurp power from the shah by naming the Minister of War and the Chief of Staff.
He did no such thing. He was a popular national figure who was democratically elected to office. Well, as democratically as Iran was at the time, as you'll see in the link. This resulted in instability, as the Shah didn't want to give up power and the allies didn't want to give up their influence.
He pushed for many reforms in Iran, one of which was to nationalize Iran's oil industry. That would have had a major impact on British and American influence in the area, and the US couldn't have that. Authorized by Eisenhower, the CIA instigated a coup that overthrew his government and put the Shah back in power, practically as dictator.
The CIA has acknowledged all of this. There's plenty from a search here.
Before anyone asks, none of this justifies Iran's attacks on Israel or should be used as arguments against Israel retaliating.