There are also semi-presidential republics, which function differently. In those systems, the US-style president role is split into two different roles, president & prime minister. President handles foreign policy, the army and selects the prime minister with the approval of parliament, and the prime minister handles everything domestic. This separation of roles means the amount of damage an individual can do is much more limited.

Edit: Oh, I missed somebody already talked about the French system, which is a semi-presidential system. Oh well, leaving this up for posterity.