Vaishnavism or Vishnuism

In Vaishnavism or Vishnuism, Vishnu Bhagwan is considered the Supreme God. The followers of Bhagwan Vishnu and His incarnations are called Vaishnavas. Vishnu has several incarnations. Rama and Krishna are the most prominent of his incarnations, and they are worshipped as veritable Narayana & as personal God. Vaishnavism is the most prominent faction within Hinduism as followed by majority of Hindus. The other factions are Shaivism, Shaktism, and Smartism.
 
Hindu philosophers such as Ramanujacharya, Nimbakacharya, Madhvacharya, Vallbhacharya, Ramananda and Sri Chaitanya Mahaprabhu were the fulcrum of the Bhakti movement during medieval period and rise of Vaishnavism.  They established various Sects & Sampradayas which spread worship of Sri Vishnu, Sri Krishna and Sri Rama along with Mata Lakshmi; Sri Radha; and Sita (also Laxman, Hanuman) respectively.
 
The worship under Vaishnavism involves ritual image worship and matra-japa. Vaishnav festivals such as Ramanavmi, Srikrishna janamashtami, Vaikuntha Ekadashi etc. are very popular.
 
Roots & Scriptures:
The Vedic deities Nara and Narayana form one of the Vedic roots of Vaishnavism. There are references which make out that Narayana is prior to the phase of adoration of Brahma, Vishnu and Siva as Gods. Vishnu Bhagwan finds mention in Rigveda as solar deity (Aditya).  Later on, Narayana got slowly identified with Vishnu.
 
Of the 108 Upanishads, fourteen are considered Vaishnava Upanishads. These are the Narayana Nrsimhatapani, Mahanaryana, Ramarahasya, Ramatapani, Vasudeva, Avyakta, Tarasara, Gopalatapani, Krsna, Haygriva, Garuda and Kali-Santarana Upanishads. The Epics attempted to make God's relation to man more personal, so that the human heart may yearn for Him. The two Indian epics, the Mahabharata and the Ramayana which presented Vaishnava philosophy, played pivotal role in spread of Vaishnavism. The philosophy and teachings of the Vaishnava Puranas ( mainly, Bhagavata Purana, Vishnu Purana, Naradeya Purana, Garuda Purana, Vayu Purana and Varaha Purana) are bhakti oriented. These puranas provided further impetus to Vaishnavism.
 
Emergence & Spread:
The ancient emergence of Vaishnavism is unclear, the evidence inconsistent and scanty.

1. During Gupta dynasty (4th to 7th centuries), Vaishnavism was adopted as the royal religion. The Gupta Empire flourished in northern, western and southern India.
2. The seed of Vaishnavism in South India was sown by the 12 Alvars, who were the earliest Brahmin messengers to the South. They gave prominence to the emotional side of Vaishnavism and used the language of the people in their songs and hymns mainly based on life of Sri Krishna. Alvers spread bhakti movement in southern region under the Pallavas and Pandyas in the 2nd to 8th centuries.  Their lives and compositions comprehend all forms of personalized love for God with attitudes known as Dasya (servitude), Apatya (sonship), Sankhya (comradeship), Vatsalya (affection for a child) and Madhura (conjugal). Nethamuni collected and edited all the devotional hymns of Alvers into a text of four thousand verses (Nalayira Divya Prabandha) which came to be recognized as the Tamil Veda.
3. With the outgrowth of the bhakti movement there was proliferation of devotional literature in vernacular prose and poetry in a number of other ethnic languages of the various Indian states or provinces. In the period between the 14th-17th centuries, the bhakti movement spread northward into Muslim-dominated India due to the efforts of a loosely associated group of Vaishnavite teachers including Chaitanya Mahaprabhu (West Bengal), Vallabhacharya (UP & Rajasthan), Mirabai (Rajasthan & Gujarat), Kabir (UP), Tulsidas (UP), Tukaram and several others.

Sects & Sampradayas:
Vaishnavas are typically divided into two major (though not mutually exclusive) sects: Bhagavatas and Pancharatras. Pancharatras Tantra by Narada, and Bhagavadgita are the foundation of the Bhagavata school.
 
A. Bhagavatas:
Bhagavatas follow Vasudeva-Krishna-Vishnu worship, dating back as far as 115 B.C. Within Bhagavatism, there are four main sampradayas, which follow different philosophical systems regarding the relationship between the soul (jiva) and God (Vishnu).
 

1. Lakshmi-sampradaya or Srivaishnavism – This was founded in 12th century by Ramanujacharya who propounded Visishtadvaita or qualified non-dualism. Vishnu bestows moksha upon the devotee by his grace, though it is through Vishnu's consort Sri (Lakshmi) that grace is bestowed.

 
The Srivaishnava tradition had bifurcated into two branches, the northern Vadagali group and the Tengali group by 14th century. The Vadagali group believes that one must cling to god much like an infant monkey clings to its mother—that is, by putting forth effort to receive god's grace. The Tengali group believes that one must put forth no effort to acquire god's grace—like a kitten, they must rely entirely on the mother-cat to pick them up and carry them to safety.
 

• Rudra-sampradaya – This was founded in 13th century by the dualist philosopher Visnusvamin. Vallabhacharya (1479-1532) expounded the philosophy of Shuddhadvaita ("pure nondualism"). Vallabhacharya decried role of ritualism, and instead claimed that all spiritual actions should be characterized by a sense of playful effortlessness. This inspired the raslila dramatic performances.
• Brahma-sampradaya was founded by Madhvacharya in 12-13th century and was based almost solely upon the Bhagavad Purana. Madhva propounded the philosophy of Dvaita. Vaishnava Bhakti movement in Karnataka like Purandara Dasa, Kanaka Dasa, Raghavendra Swami, and Chaitanya Mahaprabhu were influenced by Madhva's dualist teachings.
• Sanakadi-sampradaya was born in Pandarpur in Andhra Pradesh who founded by Nimbarkacharya (died in 1162, some maintains his birth was in 5000 BC), a Telegu brahman. He propounded the philosophy of dvaitadvaita. Nimbarka's teachings were on the concept of prapatti or "surrender" Devotees must fully surrender to the grace of god.
• Gaudiya Vaishnavism was founded by Chaitanya Mahaprabhu in 15th century in Bengal. It was marked by fervent love for the personalized god Krishna, sublimated in the form of sankirtana: public dancing, chanting, singing and ranting upon the god-man as well as his consort Radha and his incarnations. This tradition was expanded beyond India due to the efforts of Swami Prabhupada, who founded International Society for Krishna Consciousness (ISKCON).
• Ramanandi movement is named after its main proponent, Ramananda (Born in Prayagraj  UP and settled in Varanasi, lived in 1300-1411 century), Ramananda, who was the successor of Ramanuja, placed devotional focus upon Rama, whom he considered supreme Lord, and his wife Sita. Ramananda used vernacular language for the composition of his hymns. He was the spiritual stimulator & Guru of the twelve great leaders of Vaisnava cult in North India. They belonged to all caste. Among the group's most prominent members were Tulsidas, Kabir (Muslim), Mirabai, Ravidas (cobbler by birth), Sena (barber by caste), Dana (jat), Pipa (Rajput), Dadu and others.

 
B. Pancharatras:
Pancharatras focused upon the worship of Narayana. According to the Pancharatra doctrine, God is manifest in five forms. These are called Para or the supreme form of His transcendent being; Vyuha or the group of His forms called Vasudeva (cosmic consciousness or highest self), Samkarshana (individual self), Pradyumna (mind) and Aniruddha (egoism); Vibhava or His glory seen through His incarnations or Avataras; Archa or His presence manifest in His idols and images worshipped by devotees; and Antaryamin or His immanent presence within the Universe.
 
This teaching is based upon Samkhya philosophy whereby Vasudeva is the supreme purusha which gives rise to the Samkara when brought into contact with the material (prakriti) body. The Samkara is responsible for the production of the Pradyumna, and from the Pradyumna comes the Aniruddha, the creative agent. Devotees emphasize upon visual representations of Vishnu and his various incarnations and cultivating devotion via reverence for these images.
 
Sri Vaishnava Sampradaya or Sri Vaishnavism was founded by Nathamuni in 10th century in Tamil Nadu. Sri refers to goddess Lakshmi. Lakshmi-Vishnu is worshipped in this tradition. Ramanujacharya became the most influential leader of Sri Vaishnava. The tradition traces its roots to the ancient Vedas and Pancharatra texts and popularized by the Alvars with their Divya Prabandhams. The devotional practices of Sri Vaishnava have two aspects – Bhakti and Prapatti. Bhakti is Upasana or continuous concentration on the divine with the knowledge that one is a spiritual monad forming a servant (Sesa) of God ending in its final stage in Prapatti or complete self-surrender. In Sri Vaishnava tradition, Narayana is described as the father who stands for justice and Sri, the consort, as the mother whose love extends even to the most deserving.
 
The tradition split into two sub-traditions around the 16th-century called the Vadakalai (sect giving Veda the first preference) and Thenkalai (sect giving Divya Prabandham the first preference). The most striking difference between Srivaishnavas and other Vaishnava groups lies in their interpretation of Vedas. While other Vaishnava groups interpret Vedic deities like Indra, Savitar, Bhaga, Rudra, etc. to be same as their Puranic counterparts, Sri Vaishnavas consider these to be different names/roles/forms of Lord Narayana citing solid reasons thus claiming that the entire Veda is dedicated for Vishnu worship alone. Sri-Vaishnavas have remodelled Pancharatra homas like Sudarshana homa, etc. to include Vedic Suktas like Rudram in them, thus giving them a Vedic outlook.
  
Reference:

1. Bhakti Schools of Vedanta – by Swami Tapasyananda, Sri Ramakrishna Math, Chennai.
2. https://en.wikipedia.org/wiki/Sri_Vaishnavism