What are some events that show history is repeating itself?
This depends on your point of view about various events. But the one I'll offer is the war in Afghanistan as a parallel to the Vietnam War.
What is similar here is that the United States thought in both cases that it could go in to a country that had no real history of democracy and no reason to love the US and turn that country into a pro-US democracy. In both cases, we were trying to do this through our military -- making them fight a war while, at the same time, expecting them to win "hearts and minds."
There are certainly differences between the two, but I think this is a fairly significant similarity and that the two are pretty comparable.
There are many good examples from history that show events repeating themselves. For instance, the recent recession seen in many Western countries is comparable in cause (low market regulation, high speculation) and effect (unemployment, collapsed property bubble) to the Great Depression of the 1930s. However, the massive increase in the size of government has meant that the situation is not nearly as drastic.
The current war in Afghanisatan harks back to the attempted Soviet invasion of the country (1979-89) and the scramble seen over the country between the Russian and British Empires in the C19th.
The failure of the UN to deal with countries such as Zimbabwe are reminiscent of similar failures in Uganda, South Africa and Zaire/Congo in the twentieth century.