Constitutional republic

A constitutional republic is a state where the chief executive and representatives are elected, and the rules are set down in a written constitution. 
The head of state and other representatives are elected but they do not have uncontrolled power. What their power is limited to is written in the constitution. If there is dispute about what the constitution means, this is decided by a court system that is independent from the representatives.
Constitutional republics usually have a separation of powers. The separation of powers means that no single officeholder gets unlimited power. John Adams said that a constitutional republic was "a government of laws, and not of men".
Supporters of such a republic argue that it is meant to be a safeguard against tyranny. No office holder can get to a position of absolute power. However, some have argued that a constitution can be written in such a way that it lets tyranny arise, and that a constitution is therefore not a fail proof safeguard against tyranny.
Aristotle was the first to write about the idea in his works on politics.
Constitutional monarchies are a special case: even though the monarch is not elected, the people still elect other governing bodies. The constitution also limits the power of the monarch.
Comparisons.
Comparisons can be made. There are countries which nominally are constitutional republics, but which have become (in effect) dictatorships. There are also countries which have no written constitution, and are monarchies, but which act in a similar way to constitutional republics.