Buddhism breaks into three overarching traditions:
Theravada— the dominant form of Buddhism in Southeast Asia, claiming to be the original form of Buddhism: The “way of the elders,” also known as Hinayana (“lesser vehicle”), Theravada encompasses all the Buddhist traditions that are based on the Pali scriptures, recorded (according to tradition) by Shakyamuni Buddha’s disciples. Theravada Buddhist schools are dominant in Sri Lanka, Burma, Thailand, and Cambodia. In the West, there are traditional Theravada temples in Southeast Asian communities, as well as some founded by Western students of Theravada teachers. Theravada is best known in the west through Vipassana or Insight Meditation centers founded by Western students of Theravada teachers.
Mahayana— the dominant form of Buddhism in Northeast Asia, claiming to be a more advanced form than Theravada: The “great vehicle,” a general name for a huge variety of schools that were formed during a doctrinal split in the 1st century CE. All Mahayana traditions are rooted in the Prajnaparamita sutras and the Yogacara and Madhyamaka philosophical texts, which stress the emptiness of all dharmas, including all mental phenomena. Mahayana schools include Pure Land, Tiantai (Tendai), Huayan, and Zen, which are dominant throughout East Asia (China, Korea, Japan, and Vietnam). Apart from Zen, other Mahayana schools active in the West include traditional Pure Land temples and Soka Gakkai.
Vajrayana— the dominant form of Buddhism in Tibet and Mongolia claiming to subsume both Theravada and Mahayana: The “diamond vehicle,” another general term for all the forms of tantric Buddhism practiced in Tibet (and the Tibetan diaspora), Mongolia, and Central Asia. Vajrayana Buddhism preserves not only the tantric practices that arose in the last centuries of Indian Buddhism, but also a huge variety of philosophical schools based on later variations of Mahayana Buddhism. There are hundreds of Vajrayana schools and lineages active in the West, including the Gelug (the Dalai Lama’s sect), Kagyu (for example, Chogyam Trungpa’s Shambhala school), Nyingma, and so on.
The following terms and links are foundational Buddhist teachings you should be familiar with. The links provide concise definitions, definitions and further references. (Zen Master Seung Sahn’s The Compass of Zen is an excellent reference for most of them).
Four noble truths
Three marks of existence: impermanence, suffering, non-self
Dependent origination (Pratitya-samutpada)
Mind only (yogacara)
Bodhisattvas and the Bodhisattva Path
Skillful means (Upaya)
Other texts that can provide grounding in the fundamentals include:
Karen Armstrong, Buddha
An excellent contemporary life of Shakyamuni Buddha, the historical Buddha, Siddhartha Gautama.
Bhikkhu Bodhi, In the Buddha’s Words
An accessible introduction to the Theravada tradition and the Buddha’s earliest teachings.
Wahpola Rahula, What The Buddha Taught
A classic introduction to the Theravada tradition.
Lama Yeshe, Introduction to Tantra
For students interested in Tibetan Buddhism, this is probably the single best text to start with, and a helpful corrective for many misperceptions about tantra.
Princeton Dictionary of Buddhism
An essential reference, published recently, that covers all aspects of the Buddhist tradition.