Why do some countries have both a president and a prime minister?

Question

For instance, Russia has both.

Answer 1

So they can watch each other.  

Answer 2

In those countries, the President usually serves as a ceremonial head of state rather than the head of the government.

Answer 3

Mr Stiggo is correct.  The prime minister usually has the power.  The president is ceremonial.  In that regard, the president of a republic serves some of the functions that the monarch of a constitutional monarchy does--greets visiting heads of state, raises money for charity, opens schools, bridges, and so on.

Answer 4

The simple answer is this, the definition and duties of President and Prime Minister, as well as king, or whatever other title is not universal.