re·pub·lic
rəˈpəblik
noun
a state in which supreme power is held by the people and their elected representatives, and which has an elected or nominated president rather than a monarch.
archaic
a group with a certain equality between its members.
de·moc·ra·cy
dəˈmäkrəsē
noun
a system of government by the whole population or all the eligible members of a state, typically through elected representatives.
1a : government by the people; especially : rule of the majority b : a government in which the supreme power is vested in the people and exercised by them directly or indirectly through a system of representation usually involving periodically held free elections.
As you can see they are both essentially the same except that a republic has a president.