Some little basics for Caliph Era mostly asked in Viva-Voce:

1- Abu Bakr served as caliph from 632 until his death in 634. His first major accomplishment was to deal with the problem of the Bedouins (nomadic Arabs). Although some had converted under Muhammad, after his death they rejected Islam and refused to obey Abu Bakr. In 633, the caliph defeated the Bedouin revolt, known as the Ridda, and thereby secured the entire Arabian peninsula for Islam. The experience served to convince Abu Bakr that Islam needed to expand beyond Arabia in order to be secure. He set his sights on the two neighboring empires he viewed as threats to Islam: the Sassanid Empire to the east in Persia and Iraq, and the Byzantine Empire to the west in Europe, Syria, Egypt, and the Mediterranean Sea. He declared a jihad against the Byzantine Christians, but died before he was able to carry it out.

2- The second caliph was Umar, another father-in-law of Muhammad, who had been named by Bakr as his successor. His caliphate lasted from 634 to 644. One of his first contributions was to add "Commander of the Faithful" to his title, which was used by all subsequent caliphs. His primary contribution, though, was a series of military victories resulting in the rapid expansion of Islam. He conquered Damascus in 635 and Jerusalem in 637, both from Syria in the Byzantine Empire. Realizing the importance of loyalty in his new subjects, Umar instituted a policy of religious tolerance in his new lands. This was received gratefully by Jews and Christians, who had been persecuted under the Byzantines. He instituted two taxes, the kharaj for landowners with productive fields and the jizya, which non-Muslims paid in return for the privilege of practicing their religion.
At the same time, Muslim forces were moving against the Sassanid Empire in the east. Once he had secured his place in Syria, Umar succeeded in conquering the Sassanid capital, Ctesiphon, in 637. Turning west yet again, with a Muslim Syria assisting, Umar's forces set out for Egypt. Babylon fell in 641, and Alexandria in 642. Christians have not ruled in Egypt since. Umar continued the policy of tolerance in the newly-conquered lands, and Muslims did not force conversion to Islam. They depended too much on the revenue from the jizya tax and the nonresistance of the outnumbering non-Muslims.
Muslims would find that it was not as easy to placate Persia as other conquered lands. By the time Islam arrived, the Persians had become a fiercely nationalistic people. They had their own national religion, Zoroastrianism, and considered the invading Arab Muslims inferior. Caliph Umar, Commander of the Faithful, was assassinated by a Persian Christian in 644. But by the time of Umar's death, the Muslim Empire was second only to the Chinese Empire in size.

3- USMAN, a member of the influential Umayyad family, was chosen as Umar's successor, leaving Ali's supporters once again disappointed and angry. USMAN served as the third caliph from 644 to 656. In 645, he defeated a Byzantine attempt to recover Alexandria, and in 647 he began expanding the Muslim Empire west of Egypt. He conquered Cyprus in 649 and his forces reached the easternmost boundary of Persia in 653.
Some of USMAN's other accomplishments, however, were not as popular among Muslims. He appointed fellow members of the Umayyad family to administrative positions, depleted the treasury with his lavish spending habits and lack of financial planning, and perhaps most controversial of all, he sought to create a single, definitive text of the Qur'an. He succeeded in accomplishing his goal, and thereby significantly reduced doctrinal disagreements, but not without criticism from those who suspected USMAN of tampering with the sacred texts. In any case, USMAN's compilation of the Qur'an must certainly be considered a significant accomplishment for Islam.

4- Discontent abounded in the new empire. In 656 USMAN was assassinated in his home by a group of Egyptians, and civil war immediately erupted. Muslim fought Muslim over who would next assume leadership. The never-resolved conflict between Ali's supporters and other Muslims came to a head. Ali declared himself the fourth caliph, a claim which was promptly challenged by Mu'awiya, USMAN's cousin and the governor of Syria. At the "Battle of the Camel" in December 656, Ali's forces killed two of Muhammad's friends and kidnapped one of his widows.
Before long, a strong public outcry against the violence led Ali and Mu'awiya to agree to submit to the decision of a council, which would use the Qur'an as a guide in deciding who should be caliph. But when the council concluded that both should step down, Ali refused, and civil war continued. It was at this point that another another division arose within Islam. The Kharijites, a group of Shi'ites and supporters of Ali, were angry at his ever agreeing to submit to a human decision on a matter that should only be decided by God. Refusing allegiance to both Ali and Mu'awiya, the Kharijites appointed their own caliph.
In July 660, Mu'awiya declared himself caliph in Jerusalem. He had on his side not only Egypt and Syrian forces, but the Kharijites as well. The latter, intending to kill both Ali and Mu'awiya, got to Ali first. With Ali out of the picture, Mu'awiya was finally successful in claiming control of the Islamic Empire. The civil war came to an end, and the Umayyad Dynasty began.