Rather simplified, somewhat caricatured presentation based on my own very limited knowledge:
Nyingma - "the old (translation) tradition", based on prehistoric transmissions of B into Tibet as early as 8th century. Characterized by loose "democratic" organization (until the exile each monastery was independent, with no central figurehead a-la Dalai Lama) and by a very abstract main teaching intended to plant the student into an already-enlightened perspective. Historically situated in Eastern Tibet. Before the communist invasion this was a very deep, very conservative school, very proud of its archaic methods.
Kagyu - "the oral linage" from mostly Eastern (with some important monasteries in Central) Tibet, from 12th century. Characterized by emphasis on effort and various transformative / cleansing practices, this school takes pride in live transmission of fundamental sanity and teaching by personal example that is not shy of adapting to the spirit of times.
Sakya - the study school, historically in close affinity with rich secular rulers of Central and Eastern Tibet since 13-14cc. Emphasizes thorough knowledge of Dharma through study of texts. Not always considered a mature monastic tradition by the above two schools.
The head role is passed down the family line, as in hereditary monarchy.
Gelug - the new reform school, founded in Central Tibet in 14th century, on the textual basis of the then extinct Kadam school (11-12cc). The doctrinal foundation of the Central Tibetan government that saw itself in charge of Tibet from 15th to 20th century. Characterized by relatively more rigid orgstructure. Emphasizes logic and study of Madhyamaka. The Gelug is the most recent and perforce drew upon the prior schools (Nyingma/Kagyu/Sakya) in many ways.
Bon - an ancient shamanic tradition germinated by influx of Buddhism, specifically Nyingma teachings. Leads with shamanism as upaya in lower yanas, elevating to sutra-, tantra-, and Dzogchen-level teachings depending on student's capacity.
Ri-me - a universalistic movement in Tibetan Buddhism intended to harmonize the various views and styles of the different traditions while preserving their individual features. Voiced by leading Nyingma, Kagyu and Sakya scholars of 19th century, Ri-me re-established the non-sectarian principle that had been always present in Tibetan Buddhism despite the ongoing political struggle between the schools.