Obviously, it is impossible to cover a topic that is as vast as Tibetan Buddhism, in a forum answer. Very broadly speaking, I would say that Tibetan Buddhism, also known as Vajrayana and Tantric Buddhism, developed in the Himalayas; it is one of the most popular lines of Buddhism in the West. Most of its followers are from Mongolia, Bhutan, Ladakh, and Tibet.
There are also important communities in the Indian state Sikkim, Nepal, and in regions such as Buryatia, Kalmykia, Tuva, and Inner Mongolia in China. It is the religion of the Tibetan and Mongolian people who see the Dalai Lama as the Bodhisattva, therefore Tibetan Buddhism is also called Lamaism. Within Tibetan Buddhism, Tibetan monks take an important place, both religiously and socially.
Tibetan Buddhism comprises four main lineages and other minor ones. Great importance is given to the lineage because it guarantees the connection to Buddha’s original message.
These lineages are:
1) Nyingma School: "Nigma"( Red School or “of the red caps”) was founded in the 11th century and is the oldest among Buddhist sects in Tibet. Its name derives from the color of the monks’ clothing. The master Padmasambhava was the first, according to the Tibetan tradition, who controlled nature’s deities of Tibet and other forces, making Buddhism the official religion there. This school has followers mainly in areas inhabited by Tibetans in China, India, Bhutan, Nepal, Belgium, Greece, France, and United States.
(2) Sagya School: "Sakya" (striped school) was founded in 1073 with the name Sagya (white clay in Tibetan). The name comes from that of their main monastery called Sagya, founded by Konchok Gyalpo, which has white walls. It is also called the Striped Sect because the walls of their other monasteries are painted with red, black, and white stripes which represent, respectively, the Buddha of wisdom, the Goddess of mercy and guardian spirits. His main teachers are descendants of the disciples of Padmasambhava, and Shantarakshita.
(3) Gagyu School: "Kagyü" (White School) was founded in the 11th century; this sect gives high importance to Tantric practice and imparts its wisdom through an oral tradition rather than through literature. For this reason, Gagyu means oral transmission. This line was founded by a disciple of Milarepa, called Gampopa, and from the teachings of the Hindu mahasiddhas Tilopa and Naropa, which were carried to Tibet by Marpa. The Kagyu School of Karma Kagyu is directed by H.H. Karmapa and constitutes the main school of Bhutan.
Kagyü is one of the four main lineages of the Mahayana Buddhism. In addition, other subdivisions exist within this sect, such as Phaktru, Tsalpa, Barom, Shangpa, and Karma Kagyu School. Furthermore, the Phaktru line includes the Drikung, Drukpa, Taktung, Yasan, Trophu, Shuksep, Yelpa, and Mertsang lines.
(4) Gelug School: "Gelupa" (Yellow School) was founded in 1409, having emerged from the other sects as a spiritual reform of Lama Tsongkhapa. Its top leader is the Dalai Lama. It has six important monasteries: Zhaibung, Sera, Tashilhungpo, Tar, Labrang, and Gaindan. It has a reputation for its reincarnation system of the Dalai and the Panchen.