As mentioned above, the Theravada tradition is based on the set of teachings decided by the Third Council to contain the teachings of the Buddha.

Shri Lanka has played a central role in preserving the Theravada scriptures and practices. After the Third Council, the Tripitaka collection of sutras were taken to Shri Lanka. Most of these were originally in the Pali language, but some were compiled in other languages. Through the centuries however, all teachings were translated into Pali (around 35 BCE). Initially, most ordained Sangha were known as parivrajahas (wanderers). They would assemble during the rainy season when travelling became problematic. Gradually, buildings were donated and the Sangha became more static. Just a century after the Buddha passed away, monasteries became the main mechanism for preservation of the teachings. Also extra monastic rules were introduced. Only during one short period in history Buddhism was banned in Shri Lanka, but it was later restored with teachings from Thailand which in turn had originated in Shri Lanka. The main countries where the Theravada tradition is currently alive and well in Shri Lanka, Thailand, Burma, Cambodia and Laos.
The teachings on the Four Noble Truths and meditation form the basis of Theravada practice.

The term Hinayana (smaller Vehicle) appeared only much later, around the first century CE, when teachings of a different nature appeared which were called Mahayana (greater Vehicle).
In India, non-Mahayana or Hinayana sects developed independent from the form of Buddhism existing in Sri Lanka. Today, there is no Hinayana tradition in existence anywhere, although Theravada could be called the tradition most like Hinayana. The ultimate goal of the Theravadin and other non-Mahayana practice is to attain the state of an Arhat, as Buddhahood is considered practically unachievable for nearly everyone within this aeon.
Although helping other sentient beings is accepted as an important Buddhist practice, the main motivation for following the spiritual path is to achieve liberation for oneself - Nirvana.
Due to the negative connotation of the term Hinayana, the World Fellowship of Buddhists decided that the term Hinayana should be dropped to refer to Buddhism existing today, and the term Theravada should be applied, also because the term Hinayana has a negative connotation.

MAHAYANA

The Mahayana appears to have developed between the 1st Century BC to the 1st Century CE. About the 2nd Century CE Mahayana became clearly defined. Master Nagarjuna developed the Mahayana philosophy of Sunyata (emptiness) and proved that everything is 'Void' (not only the self) in a small text called Madhyamika-karika. After the 1st Century CE., the Mahayanists took a definite stand and only then the terms of Mahayana and Hinayana were introduced.

Around the first century CE, teachings of a different style appeared. The terms Mahayana and Hinayana appeared in the Saddharma Pundarika Sutra or the Sutra of the Lotus of the Good Law. Of great influence to the development of the Mahayana was Master Nagarjuna (2nd Century CE) who is known for his profound teachings on the philosophy of emptiness. About the 4th Century CE, the Masters Asanga and Vasubandhu wrote enormous amount of works on Mahayana. The Mahayana teachings were mainly written down in Sanskrit, and are now called the Mahayana Sutras.
A clear division arose between the schools following the traditional teachings and Mahayana. Although the main philosophical differences may be small, they have profound consequences for the practices involved.
The Mahayana philosophy is based on the older tradition and fully accepts these teachings, but not all traditional interpretations. One of the most important aspects is for example the traditional interpretation that Buddhahood can be achieved only by very few people. The Mahayana teaches instead that every sentient being (being with a mind) can become a Buddha, the only thing preventing our full enlightenment is the failure to improve one's own actions and state of mind. The Mahayana tradition claims that all their sutras have been taught directly by Shakyamuni Buddha or have at least been inspired by the Buddha.
The main Mahayana motivation is to lead all sentient beings to enlightenment. Liberation from cyclic existence (Nirvana) and Buddhahood for oneself are regarded simply as fortunate by-products of one's efforts to help all beings. In fact, the only possible motivation with which one can become a Buddha is the altruistic wish to lead all sentient beings away from suffering.
This motivation is reflected in taking an additional set of vows, known as Bodhisattva vows on top of taking Refuge. The main vow is to free all sentient beings from suffering. These vows are not taken for this life only, but for all future lives as well, until this goal is achieved. The main practices of a Mahayanist are summarised in the 6 perfections: the perfection of giving, ethics, patience, joyous effort, concentration and wisdom.
The Mahayana tradition mainly developed in North India, and spread further North into China and Tibet. In China, Buddhist philosophy and practice was often mixed with Taoist and Confucian aspects. Via China, Mahayana Buddhism also spread to other countries like Korea, Vietnam, Cambodia, Laos and Japan. Also, in China the Ch'an tradition evolved, which was introduced into Japan, and there developed into Zen. Also, the very popular Pure Land Buddhism developed, which focuses on being reborn in the Pure Land of Amitabha, mainly through recitation of Amitabha Buddha's name. Pure Land Buddhism is known as Jodo in Japan.

In contrast to the current very clear division between Theravada and Mahayana schools, it must be noted that for many centuries, monasteries in India were filled with monks of both traditions. It was considered a very personal decision to choose for individual liberation or Buddhahood. The monastic and ordination rules are virtually the same, and the teachings overlap to a great extent. See for example this important text from the World Buddhist Sangha Council convened by Theravadins in Sri Lanka in 1966.

TANTRAYANA

Around the 6th. century AD, within the Mahayana tradition the tantras or tantric texts emerged. Based firmly on the Hinayana and Mahayana tradition, the actual philosophy differs only slightly from the Mahayana, but the practices can be quite different.
Prior to engaging in tantric practices, a proper understanding of the Hinayana and Mahayana philosophy is considered essential. Only then should one obtain initiation or permission from a qualified tantric master to do a specific tantric practice.
Tantric practices are psychologically very profound techniques to quickly achieve Buddhahood. This is considered important, not for oneself, but because as a Buddha one has the best achievable qualities to help others. The motivation is: 'the faster I can achieve Buddhahood, the sooner I can be of maximum benefit to others'.
Depending on the class of tantra, extra vows may need to be taken on top of the Refuge and Bodhisattva vows. Also, specific commitments may be required like doing a specific retreat, daily recitation of mantras or a daily meditation practice. (For more details see the page on Tantra.)
In the 8th. century, the Mahayana and Tantrayana (or Vajrayana) traditions of (North) Indian Buddhism were introduced into Tibet. In fact, only in Tibet, Bhutan and Mongolia a virtually complete set of tantric teachings was preserved. The Tibetan tradition can also be found in the Himalayan range of Ladakh (Northwest India), Sikkhim (Northeast India) and Nepal, and in Mongolia (which is virtually identical to the Tibetan tradition). In China and countries like Korea and Japan, remnants of Vajrayana can be found.

The term Sutrayana is used within the Mahayana to indicate the non-tantric Mahayana teachings.

DO THESE TRADITIONS CONTRADICT?

His Holiness the Dalai Lama noted the following in the book 'The Heart Sutra':

"It is very important to understand that the core teachings of the Theravada tradition embodied in the Pali scriptures are the foundation of the Buddha's teachings. Beginning with these teachings, one can then draw on the insights contained in the detailed explanations of the Sanskrit Mahayana tradition. Finally, integrating techniques and perspectives from the Vajrayana texts can further enhance one's understanding. But without a foundation in the core teachings embodied in the Pali tradition, simply proclaiming oneself a follower of the Mahayana is meaningless.
If one has this kind of deeper understanding of various scri-tures and their interpretation, one is spared from harboring mis-taken notions of conflicts between the "Greater" versus the "Lesser" Vehicle (Hinayana). Sometimes there is a regrettable tendency on the part of certain followers of the Mahayana to disparage the teachings of the Theravada, claiming that they are the teachings of the Lesser Vehicle, and thereby not suited to one's own personal practice. Similarly, on the part of followers of the Pali tradition, there is sometimes a tendency to reject the validity of the Mahayana teachings, claiming they are not actually the Buddha's teachings.
As we move into our examination of the Heart Sutra, what is important is to understand deeply how these traditions complement each other and to see how, at the individual level, each of us can integrate all these core teachings into our personal practice."