The early community divided over discipline and doctrine long before those differences hardened into the streams recognised today, and the conventional threefold division flattens a great deal.
Theravada, the way of the elders, is the surviving school of the early period and predominates in Sri Lanka, Myanmar, Thailand, Cambodia and Laos. It takes the Pali canon as authoritative, holds up the arhat, one who has ended the defilements and will not be reborn, as the goal, and gives the ordained community a central place.
Mahayana texts began to appear around the turn of the Common Era, presenting themselves as teachings of the Buddha not previously disclosed. The change is in the goal: the bodhisattva works for the liberation of all beings rather than for an individual release, and buddhahood rather than arhatship becomes what practice aims at. Mahayana thought developed the analysis of shunyata, emptiness, systematised by Nagarjuna, and the Yogacara account of mind. It became the dominant form in China, Korea, Japan and Vietnam, where it produced the Chan or Zen schools of meditative practice and the Pure Land traditions of devotion to Amitabha.
Vajrayana, drawing on tantric texts and methods from around the seventh century, uses ritual, visualisation, mantra and a close relationship with a teacher as means to accelerate a path it otherwise shares with Mahayana. It prevails in Tibet, Mongolia, Bhutan and the Himalayan regions, and is present in Japan as Shingon.
The relationship between the three is itself disputed. Theravada does not accept the Mahayana scriptures as the word of the Buddha; Mahayana traditions have historically characterised the earlier schools in terms those schools reject. Vajrayana is classed by many as a branch of Mahayana rather than a third stream at all.