Old and new
Nyingma means the ancient ones. It traces itself to the first diffusion, to Padmasambhava, and to the Dzogchen teachings on the nature of mind. The other three come from the second diffusion.
Kagyu descends from Marpa the translator and his student Milarepa, the hermit and poet, and was given monastic form by Gampopa; its central practice is mahāmudrā. Sakya was founded in 1073 at Sakya monastery and led by the Khon family, with a strong tradition of logic and scholarship. Geluk was founded on the work of Tsongkhapa, who lived from 1357 to 1419, and whose monastery of Ganden was built in 1409; it is known for rigorous study and strict monastic discipline.
Lineage, not schism
All four hold the same canon, take the same monastic vows, follow the bodhisattva path and practise tantra. They differ in which lineages they inherit, which texts they put at the centre, and how they read a handful of philosophical questions. Teachers often study in more than one school, and in the 19th century the rimé movement in eastern Tibet deliberately encouraged exactly that.
The English words school and sect both fit badly. Boundaries hardened over time, and often for political rather than doctrinal reasons.
Reincarnate teachers
Tibet developed a way of passing on authority that exists nowhere else in Buddhism: the tulku, a teacher recognised as the rebirth of his predecessor. It began with the Karmapas of the Kagyu school in the 13th century and spread widely, the Dalai Lamas being the best known line.
It solved a real problem, giving monasteries continuity without hereditary succession. It also made them political institutions, with estates, students and interests that outlasted any single life.
Why it matters
Together they form the living tradition of Tibetan Buddhism practised today.
History and tradition
What historians can say
Monastery records, the Blue Annals and dated colophons allow the founders and institutions to be placed with reasonable confidence. Much of what is said about the earliest Nyingma lineages comes from later sources and is harder to establish.
What Buddhist tradition remembers
Each school keeps a lineage prayer naming its teachers back to the Buddha, or to Vajradhara. Milarepa's hundred thousand songs, and Tsongkhapa's visions of Mañjuśrī, are remembered as part of that inheritance rather than as biography.