Lists and numbering of Buddhist councils vary between and even within schools. The numbering here is normal in Western writings.
According to the scriptures of all Buddhist schools, the first Buddhist Council was held soon after the parinirvana of the Buddha, dated by the majority of recent scholars around 400 BCE, under the patronage of king Ajatasatru with the monk Mahakasyapa presiding, at Rajagaha (now Rajgir). Its objective was to preserve the Buddha's sayings (suttas) and the monastic discipline or rules (Vinaya). The Suttas were recited by Ananda, and the Vinaya was recited by Upali. According to some sources, the Abhidhamma Pitaka, or its matika, was also included. Also the Sangha made the unanimous decision to keep all the rules of the Vinaya, even the lesser and minor rules.
According to the Macmillan Encyclopedia of Buddhism (2004),
About 100 or 110 years after the Buddha's Nibbana, a monk called Yasa, when visiting Vesālī, noticed a number of lax practices among the local monks. A list of 'ten points' is given; the most important was that the Vesālī monks, known as Vajjiputtakas, consented to accepting money. Considerable controversy erupted when Yasa refused to follow this practice. He was prosecuted by the Vajjiputtakas, and defended himself by quoting in public a number of canonical passages condemming the use of money by monastics. Wishing to settle the matter, he gathered support from monks of other regions, mainly to the west and south. A group consented to go to Vesāli to settle the matter. After considerable maneuvering, a meeting was held, attended by 700 monks. A council of eight was appointed to consider the matter. This consisted of four locals and four 'westerners'; but some of the locals had already been secretly won over to the westerners' case. Each of the ten points was referred to various canonical precedents. The committee found against the Vajjiputtaka monks. They presented this finding to the assembly, who consented unanimously. The canonical accounts end there.
Virtually all scholars agree that this second council was a historical event.
According to the Theravāda commentaries and chronicles, the Third Buddhist Council was convened by the Mauryan king Ashoka at Pātaliputra (today's Patna), under the leadership of the monk Moggaliputta Tissa. Its objective was to purify the Buddhist movement, particularly from opportunistic factions which had been attracted by the royal patronage. The king asked the suspect monks what the Buddha taught, and they claimed he taught views such as eternalism, etc., which are condemned in the canonical Brahmajala Sutta. He asked the virtuous monks, and they replied that the Buddha was a 'Teacher of Analysis' (Vibhajjavādin), an answer that was confirmed by Moggaliputta Tissa. The Council proceeded to recite the scriptures once more, adding to the canon Moggaliputta Tissa's own book, the Kathavatthu, a discussion of various dissenting Buddhist views now contained in the Theravāda Abhidhamma Pitaka.
Also, emissaries were sent to various countries in order to spread Buddhism, as far as the Greek kingdoms in the West (in particular the neighboring Greco-Bactrian Kingdom, and possibly even farther according to the inscriptions left on stone pillars by Ashoka). According to Frauwallner (Frauwallner, 1956), several of these missionaries were responsible for founding schools in various parts of India: Majjhantika was the father of the Kasmiri Sarvastivādins; Yonaka Dhammarakkhita may have been the founder of the Dharmaguptaka school; Mahādeva, sent to the Mahisa country may have been the founder of the Mahisasakas; and several teachers travelled to the Himalayas where they founded the Haimavata school, including a certain Kassapagotta, who may be connected with the Kasyapiyas. Relics of some of these monks have been excavated at Vedisa. (Willis, 2001) The most famous of the missionaries, and the main focus of interest for these Theravada histories, is Mahinda, who travelled to Sri Lanka where he founded the school we now know as Theravada.
The Theravāda's own Dipavamsa records a quite different Council called the 'Great Recital' (Mahāsangiti), which it claims was held by the reformed Vajjiputtakas following their defeat at the Second council. The Dipavamsa criticizes the Mahasangitikas (who are the same as the Mahasanghikas) for rejecting various texts as non-canonical: the [Vinaya] Parivāra; the 6 books of the Abhidhamma; the Patisambhida; the Niddesa; part of the Jatakas; and some verses. (Dipavamsa 76, 82)
The Mahāsanghika, for their part, remember things differently: they allege, in the Sāriputraparipriccha that there was an attempt to unduly expand the old Vinaya. The Mahasanghikas' own vinaya gives essentially the same account of the Second Council as the others, i.e. they were on the same side.
An entirely different account of Mahāsanghika origins is found in the works of the Sarvāstivāda group of schools. Vasumitra tells of a dispute in Pātaliputra at the time of Ashoka over five heretical points: that an Arahant can have nocturnal emission; that he can have doubts; that he can be taught by another; that he can lack knowledge; and that the path can be aroused by crying 'What suffering!'. These same points are discussed and condemned in Moggaliputta Tissa's Kathavatthu, but there is no mention of this Council in Theravadin sources. The later Mahavibhasa develops this story into a lurid smear campaign against the Mahasanghika founder, who it identifies as 'Mahadeva'. This version of events emphasizes the purity of the Kasmiri Sarvastivadins, who are portrayed as descended from the arahants who fled persecution due to 'Mahadeva'.
Another Fourth Buddhist Council was held in the Sarvastivada tradition, said to have been convened by the Kushan emperor Kanishka, around 100 CE at Jalandhar or in Kashmir. It is said that Kanishka gathered five hundred Bhikkhus in Kashmir, headed by Vasumitra, to systematize the Sarvastivadin Abhidharma texts, which were translated form earlier Prakrit vernacular languages (such as Gandhari in Kharosthi script) into the classical language of Sanskrit. It is said that during the council three hundred thousand verses and over nine million statements were compiled, a process which took twelve years to complete. Although the Sarvastivada are no longer extant as an independent school, its traditions were inherited by the Mahayana tradition. The late Monseigneur Professor Etienne Lamotte, an eminent Buddhologist, held that Kanishka's Council was fictitious. However, David Snellgrove, another eminent Buddhologist, considers the Theravada account of the Third Council and the Sarvastivada account of the Fourth Council "equally tendentious," illustrating the uncertain veracity of much of these histories.
As in the case of the preceding councils, its first objective was to affirm and preserve the genuine Dhamma and Vinaya. However it was unique insofar as the monks who took part in it came from eight countries. These two thousand five hundred learned Theravada monks came from Myanmar, Cambodia, India, Laos, Nepal, Sri Lanka, Thailand. The late Venerable Mahasi Sayadaw was appointed the noble task of asking the required questions about the Dhamma of the Venerable Bhadanta Vicittasarabhivamsa who answered all of them learnedly and satisfactorily. By the time this council met all the participating countries had had the Pali Tripiṭaka rendered into their native scripts, with the exception of India.
The traditional recitation of the Buddhist Scriptures took two years and the Tripiṭaka and its allied literature in all the scripts were painstakingly examined and their differences noted down and the necessary corrections made and all the versions were then collated. It was found that there was not much difference in the content of any of the texts. Finally, after the Council had officially approved them, all of the books of the Tipitaka and their commentaries were prepared for printing on modern presses and published in the Burmese script. This notable achievement was made possible through the dedicated efforts of the two thousand five hundred monks and numerous lay people. Their work came to an end on the evening of Vesak, 24 May 1956, exactly two and a half millennia after Buddha's Parinibbana, according to the traditional Theravada dating.