Who they were
When the Prophet Muhammad ﷺ died in 632 CE, leadership of the Muslim community passed to a caliph (Arabic khalifa, "successor"). In the Sunni tradition, the first four caliphs are remembered as the Rightly-Guided (Rashidun) Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan, and Ali ibn Abi Talib, who together led the community from 632 to 661 CE.567
The honor in this title reflects a saying of the Prophet ﷺ, who urged his community to hold fast to their guidance:
"You must then follow my Sunnah and that of the rightly-guided caliphs. Hold to it and cling to it firmly."
— Sunan Abi Dawud 4607 (Sahih)1
These caliphs were among the closest companions of the Prophet ﷺ, and the Quran praises the foremost of the early believers:
"And the first forerunners [in faith] among the Muhajireen and the Ansar … Allah is pleased with them and they are pleased with Him."
— Quran 9:100 (Saheeh International)2
Abu Bakr (632–634)
The first caliph, Abu Bakr, held the community together after the Prophet's death and brought the Arabian Peninsula back under unified authority. It was under him that the first effort to collect the Quran into a single written volume was undertaken, carried out by the scribe Zaid ibn Thabit:
Zaid ibn Thabit gathered the Quran "from palm stalks, thin white stones, and from the men who knew it by heart."
— Sahih al-Bukhari 49863
Umar ibn al-Khattab (634–644)
Under the second caliph, Umar, the Islamic state expanded dramatically beyond Arabia into the former Byzantine and Persian lands, and he is credited with building much of its early administration — including the adoption of the Islamic (Hijri) calendar.567
Uthman ibn Affan (644–656)
The third caliph, Uthman, oversaw continued expansion and is especially remembered for standardizing the text of the Quran: using the collection made under Abu Bakr, he had uniform copies produced and sent to the major regions so that all Muslims recited from one agreed text.
Uthman had standardized copies of the Quran written and distributed to the provinces.
— Sahih al-Bukhari 49874
Ali ibn Abi Talib (656–661)
The fourth caliph, Ali, the Prophet's cousin and son-in-law, led during a period of serious internal conflict (known as the First Fitna). His caliphate ended in 661 CE, which also brought the Rashidun period to a close and gave way to the Umayyad dynasty.567
A note on perspective: The questions of leadership and succession in this era are also where the Sunni and Shia traditions came to differ. In keeping with this site's approach, the account above presents the mainstream Sunni perspective; the Sunni–Shia distinction is a larger topic for a dedicated article.
In short
The Rashidun caliphs — Abu Bakr, Umar, Uthman, and Ali — guided Islam through its first crucial decades: preserving the Quran in writing, expanding far beyond Arabia, and building the institutions of a new civilization, before the era gave way to the great caliphates surveyed in A Short History of Islam.