The person acting in Muhammad's place after his death, i.e. the leader of Islam (sunni).
When Muhammad died in 632, the Muslim community faced a problem on how their community should be governed, and how leaders should be appointed. There were conflicting stories on what Muhammad had said. The tensions came already with the first appointment of a leader, a person acting in the place of the Messenger, a khalifatu r-rasuul, a few months after Muhammad's death. Khalifa can be translated with 'successor, vicegerent', but is a term that is seldom used for anything else than the leader of the entire Muslim community. But when it is used, like in honorary titling of good Muslims, it always refer to its standard meaning.
Through history, we have seen parallel Caliphs, but none had as much symbolic power and influence as the one that followed the line of Caliphs from Abu Bakr, which was the first. This line of Caliphs had a steady residence in Damascus from 661 to 750 and Baghdad and  up until 1258. After 1258 and until 1924 there have been several Caliphs, but all of these have had only limited influence. They have represented no continuation of the Caliphs of Baghdad, and in more than one case, these caliphhoods have been motivated by political motives rather than religious ones. The Muslim world has never agreed upon uniting behind anyone of these.
The Caliph carried other titles which were less modest, as they were not relative to Muhammad, but to the Muslim community. As amiiru l-mu'miniin he was responsible for the Muslim armies. As imaam he was the head of public worship, and gave khutbas.
The last Caliph was removed by the Mongols when they conquered Baghdad. After this there have been several rulers putting up their own Caliph, but the Caliphate never gained any of its former power or importance.
There are four periods of the Caliphate of Islam:

RASHIDUNS (632-661)

Rashidun is the name used for the four first Caliphs, from 632 to 661 and indicates that these were the just and admirable leaders of the Muslim community. This period was marked by a long line of conquests by the Arabs as well as endeavors to turn the leaflets of the revelations that had been given to Muhammad into a book, the Holy Koran. Inside the Muslim realms peace prevailed until the death of Uthman in 656. As this was a murder, the Muslims could not agree upon quite who was responsible. This time, the caliphate of Ali, came with the two schisms that have impregnated Islam ever since, when first there was a break between the majority and a group now known as Kharijis, and later between the group now known as Shi'is and the Sunnis.

UMMAWIYYS (661-750)

The Ummawiyys got their power through military actions, a fact that influenced their religious legitimacy strongly through the 90 years they had the power. Most Muslims regard the Ummawiyys as less admirable than both the Rashiduns and the later Abbasids. Even if the Shi'is did not accept the rule of the Ummawiyy Caliphs, this group was at the time too weak to represent much of a threat to the ruling group.

ABBASIDS (750-1258)

The Abbasids were to a large extent Shi'is (the division lines of today were not as clear in those early days) and the defeat of the Ummawiyys was strongly motivated by Ali's claim on the leading position in the Muslim world. The Abbasid Caliph involved himself strongly in the religious life of the community. The distance between ruler and people became longer, the court of the Caliph was one of increasing splendor.
The 9th century was the start of the decline of the real influence of the Caliph on first politics, and soon also religious matters. The symbolic importance was, however, increased. All effective power was lost in 946. The Buyyids became the new ruling dynasty, but in secular terms. Some cases of outward importance of the Caliph were seen in the following centuries, but these were mainly instances where the secular ruler got the blessings of the Caliph without giving the Caliph any form of influence. The blessings in the shape of a diploma of investiture and robes of honor were given to leaders as strong as Saladin.
In 928 Abdu r-Rahman III of Spain, a descendant of the Ummawiyys, took the title caliph, a title his descendants also carried. The Fatimids of Egypt had also taken this title, as far as back to 909, but they put less emphasis on this than what the Ummawiyys of Spain did.

PERIOD SINCE 1258

When al-Musta'sim was killed in 1258 by the Mongols, he did not leave any heir. The uncle of al-Musta'sim was however installed in the position as Caliph in 1261 in Cairo, but this Caliph disappeared in the desert when bringing an army up north in order to try to sack the Mongols. A new Caliph was installed in 1261, once again in Cairo, this also a relative of al-Musta'sim. A mere symbol, without the permission to move freely around, this new line of Caliphs stayed in their position for about 250 years. Except from installing the Sultan in great ceremonies, this Caliph had no importance. The Abbasid Caliph of Cairo was also ignored by the rest of the Muslim world.
In several places Caliphs popped up, in Maghreb, with the Saljuqs, the Timurids, the Turkomans, the Uzbeks and the Ottomans. When the Ottomans conquered Egypt in 1517, the remaining Caliph was transported to Constantinople, the Ottoman Sultan, Selim, also called himself Caliph. Later sources claims that the Abbasid caliph transferred his dignity to Selim. In the 18th century the importance of being Caliph had grown stronger for the Ottoman Sultan, and started to call himself the protector of the Muslim religion. Some influence did the Ottoman Caliph and Sultan have. With the fall of the Ottoman Empire, the Sultan held on to his title of Caliph for two more years, until his office was abolished in March 1924.
A congress in Cairo in 1926, that tried to reestablish the Caliphate, did not manage to succeed. Important Muslim countries did not participate, and the resolutions agreed upon did not result in real actions, even if they expressed to be in favor of a Caliphate. Since then nothing has been done, much due to nationalism in the different countries.