Short answers, pulled from the story.
Sunnī Islam is by far the largest branch, comprising 87-90% of the world's Muslim population. Shīʿa Islam is the second largest, at 10-13%. Ibadi Muslims number about 2.7 million, and a quarter of the world's Muslims identify as non-denominational, belonging to no specific branch.
The original schism arose from a dispute over who should lead the Muslim community after the death of the prophet Muhammad. Sunnīs accepted Abu Bakr as the first caliph, chosen by private election. Shīʿas believed ʿAlī ibn Abī Ṭālib, Muhammad's cousin and son-in-law, was the rightful successor. The Kharijites broke from both groups during the First Fitna, the first Islamic Civil War, following the assassination of the third caliph Uthman in 656.
The four main Sunnī schools of jurisprudence are the Ḥanafī, Mālikī, Shāfiʿī, and Ḥanbalī schools, named after their respective founders Abū Ḥanīfa al-Nuʿmān, Mālik ibn Anas, Muḥammad ibn Idrīs al-Shāfiʿī, and Aḥmad ibn Ḥanbal. All four founders lived in the 8th century CE.
Shīʿa Islam is divided into three major sects: Twelvers, Ismāʿīlīs, and Zaydīs. Twelvers are by far the largest group, estimated at 85% of all Shīʿa Muslims in 2012. Ismāʿīlīsm further split into Nizārī and Musta'lī branches, with the Musta'lī dividing again into Ḥāfiẓi and Ṭayyibi; Ṭayyibi Ismāʿīlīs, known as Bohras, are split among Dawudi, Sulaymani, and Alavi groups. Zaydīs survive today only in northern Yemen.
The Nation of Islam was founded by Wallace Fard Muhammad in Detroit in 1930. It holds that Wallace Fard Muhammad was God on earth and regards Elijah Muhammad, his successor, as the true Messenger of Allah rather than the Arabian Muhammad. Its declared aim was the spiritual, mental, social, and economic resurrection of Black men and women in America and the world.
The Ahmadiyya Movement in Islam was founded in British India in 1889 by Mirza Ghulam Ahmad of Qadian, who claimed to be the promised Messiah and a subordinate prophet to Muhammad. Because Ahmadis reject the belief that Muhammad was the last prophet, their Islamic religious practices are punishable under Ahmadi-specific laws in Pakistan's penal code, and they have been officially classified as non-Muslims.