Hinayana is a derogatory term for Southern Buddhism coined by the Mahayanists. The only sect of Southern Buddhism still practiced is Theravada, so the respectful way to talk about the three is to say Theravada, Mahayana and Vajrayana.
Theravada means the way of the elders. They only recognize the Pali Cannon. The are a subsect of a subsect of pre-sectarian Buddhism , although they might argue that they are heirs to pre-sectarian Buddhism. They were founded around 200 years after the death of the Buddha in Sri Lanka.
They tend to think of themselves as the purest form of Buddhism. Comparing the Pali Nikayas to the Chinese agamas however shows important differences in pre-sectarian Buddhism and Theravada. The focus in Theravada is on ending the suffering of the individual. It is practiced in South and Southeast Asia in countries like Sri Lanka, Burma and Thailand.
Mahayana means large vehicle. They follow cannons written in many languages usually depending on the country where the Mahayanan sect originated. These can be traced back to the Sanskrit canon.
They claim to have come from secret teachings of the Buddha. Theravadins see the Mahayana as a distortion of Buddhism that came later. Scholars say that the Mahayana was a reaction to what they saw as the neglect of the heart centered practices (the brahmaviharas) by the Southern schools. Bodhicitta (the awakened heart) is central to the teachings of the Mahayana.
The Mahayanists take the Bodhisattva vow to not enter parinirvana until all beings are saved from suffering. The means continual rebirth until the job is done. This is why they call themselves the large vehicle. They coined the term Hinayana or small vehicle to insinuate that Southern Buddhists are selfish by focusing only on the individual.
There are many Mahayana sects extant. Some sects are Zen, Tendai, Pure Land and Nichiren. It is practiced in North, East and Southeast Asia in countries like China, Korea, Vietnam and Japan.
The Vajrayana or diamond vehicle (as in indestructible) was the result of syncretism between Buddhism and Tantric Yoga which came from Hinduism. Despite what you may have heard, the sexual practices are only a small part of Tantra. The word Tantra is also an attempt at superiority. The other schools read sutras (sutta in Pali). Sutra means thread. Tantra means loom.
Vajrayana has many complicated practices and a complicated system of Bodhisattvas, Buddhas and Deities. They claim that their practices accelerate the process of becoming enlightened. Some countries where Vajrayana is found are Tibet, Nepal and Japan.
©2017 Stephen L. Martin
Photo: Interior of Ajanta chaitya hall, Cave 26 by Robert Gill (c. 1868)