Monarchy is the state where monarchy institution is functioning. Monarchy can be of two types, absolute or constitutional. The most powerful man in the state is the king and if he is minor, the state is led by one or more regents. The succession to the throne is not done by elections but by the rules of succession. Most monarchies have instituted the procedure of abdication.
Republic is the state which is usually led by a president who is elected for a period of time. Republic can be of three types: presidential, semi-presidential, or parliamentary. Republics existed even in ancient times; the most famous example is the Roman Empire.
A monarchic state can be led not only by a king, but by an emperor, sultan, emir, sheikh etc. Monarchy takes the name of kingdom, empire, sultanate, emirate etc, depending on the rank of the head of the state, but it’s the same thing.
State leading modes can be of two types, dictatorship or democracy. The word dictatorship has its origins in Latin language. In the Roman Empire, which was a republic, dictators such as Caesar, Crassus, Nero, Caligula, etc appeared for the first time. Dictatorship may be composed of one person (Stalin, Hitler, Ceausescu, etc.) or more persons (the triumvirate of the Roman Empire, the dictatorship of the colonels in Greece etc.)
The term democracy comes from ancient Greek and means leadership exercised by the people.
In ancient Greece there were several independent republics or fortresses that were ruled by the vote of majority of men (women and slaves were excluded). Until reunification of Greek islands made by Alexander the Great, also known as Alexander of Macedon, there were tribes of Ionians, Dorians, Spartans, Cretans, Hellenes, etc.
In the civilized world, kings do not lead anything; they are there just for decoration (UK, Sweden, Monaco, etc.) It is not good for someone to be leader for life. The republic is more equitable. The President is elected periodically so he is supposed to abstain from doing illegal things. Institution of monarchy is a parasitic organism.