As far as I know, Sunni believe in Caliphate, and Shia believe in Imamate. My question is that: What is the difference between Imamate and Caliphate?
Largest Shiite speakers have such a definition for Imamah:
Imamah is a public leadership in the world and in religion and life affairs 1.
Allama Helli also brought this:
leader Imamah is a public leadership in religion and life for a particular person as the representative of the Prophet 2.
Imamah has various aspects such as religious authority, the full extent of the interpretation of the Qur'an, the Prophet stated correct tradition, the extent of implementation of God, resolve disputes, judgement and other aspects which is including government.
The Caliphate is the one of the issues of the Imamah, and Imamah is the succession of the Prophet in managing the society and implementation of rules and Allah's order.
This actually has nothing to do with Shia and Sunni, nor anything with religion. It matters on the context of each word being used.
Caliph means a successor or a heir in Arabic. A Caliphate is something where one succeeds another. It can be in a family, like a son is a caliph of the father, it can be a tribal system or it can be a statehood.
You can be a disbeliever and be called a Caliph or Caliphate.
It just happens that historically, Muslims were the ones using these terms so religion was mixed into it. It actually has to do with the Arabic language and very little to do with religion.
Allah says in Surah al-Baqarah 2:30=
And when your Lord said to the angels, "Indeed, I will make upon the earth a successor (khalifat)...
In Surah al-An'am 6:165 =
And it is He who has made you successors (khalaifa) upon the earth and has raised some of you above others in degrees...
In Surah al-A'raf 7:74 =
And remember when He made you successors (kholafa) after the 'Aad and settled you in the land...
And in other ayat Allah calls Bani-Israil successors (yastakhilfakom). Allah made Prophet David a Caliph.
It is alleged that Rasulullah also said=
Bani Israil were ruled by Prophets, whenever a prophet died, another would succeed (Khalafat) his place. There will be no prophet after me, but there will be successors (Kholafa) who will increase in number... [Sahih al-Bukhari, Sahih Muslim]
So the terms "caliph" or "caliphate" are linguistic. Allah is the one who gives it to whom he wills.
You can even call USA a caliphate since one president succeeds another.
Imam is used in the same manner. Imam in Arabic means a guide or a leader or something that's measured.
Allah says in Surah al-Baqarah 2:124 =
And when Abraham was tried by his Lord with commands and he fulfilled them. Said, "Indeed, I will make you a leader (Imam) for the people...
In Surah Ya Seen 36:12 =
Indeed We shall give life to the dead, and We record that which they send before and that which they leave behind, and of all things have We taken account in a clear Book (imammobeen).
The ayah uses it in a measurable sense, taking things into account.
We know Imam of the masjid (mosque) is the person who leads the prayer.
We know Imam of the country is the leader who leads.
Now lets say we took over a country. We choose a leader and we will call him Imam. But we can't call him caliph because he hasn't succeeded anyone.
Once this Imam leaves his post and another person replaces him, then we can call the second one a caliph because he succeeded the Imam.
Therefore, it matters on the context of the usage.