Buddhism eventually spread to China, Burma, Japan, Tibet, and parts of southeast Asia. Over time, Buddhism went through many stages in different coutries.

Theraveda (or Hinayana) Buddhism - 5th century BC - 1st century AD (India)

Earlier of the two great schools of Buddhism, still prevalent in Sri Lanka, Burma, Thailand, and Cambodia, emphasizing personal salvation through one's own efforts.

Mahayana Buddhism - 1st century AD - 6th century AD (India)

The later of the two great schools of Buddhism, chiefly in China, Tibet and Japan, characterized by eclecticism and a general belief in a common search for salvation, sometimes thought to be attainable through faith alone.

Vajrayana Buddhism - 6th century AD - 11th century AD (India)

Also known as Tantric or esoteric Buddhism...characterized by the practice of mandalas, mantras, and mudras. Emphasizes more female figures, while always balancing the male and female symbolism.

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(from The Basic Teachings of Buddhism)

• Theravâda ("Teaching of the Elders") Buddhism (called "Hînayâna," the "Lesser Vehicle," by the Mahâyâna): In India, 5th century BC to 1st century AD.
  
  
  • Distinctive doctrines:
    1. The Buddha (Siddhartha Gautama, Shakyamuni) is gone, and individual practitioners must work out their salvation on their own.
    2. The Buddha was unique, and individual practitioners cannot become Buddhas, only arhats ("saints"). There will be a future Buddha, Maitreya, but not for thousands of years.
    3. Nirvân[.]a (liberation) and samsâra (the place of death and rebirth) are definitely different. Samsâra is a place of suffering to be left behind. Nirvân[.]a is a liberation that is free of death and rebirth but is beyond description and rational understanding.
    
    
    
  • Places where Theravâda spread: Theravâda Buddhism is presently practiced in Shri Lanka (Ceylon), Burma, Thailand, and Cambodia. These places preserve the Buddhist canon, the Tripit[.]aka (the "Three Baskets"), in the Pâli language. During the Theravâda period, Buddhism also spread into Afghanistan, Central Asia, and Indonesia; but all those places subsequently converted to Islâm.
  
  
  
• Mahâyâna ("Great Vehicle") Buddhism: In India, 1st century AD to 6th century.
  
  
  • Distinctive doctrines:
    1. The Gautama Buddha is not gone, and individual practitioners are not on their own. Instead, the Buddha taught the dharma out of compassion, and his compassion would prevent him from being unavailable to practitioners now. Indeed, to emulate the compassion of the Buddha, practitioners become bodhisattvas, who vow to carry all beings with them into salvation. Bodhisattvas are also available, like the Buddhas, to help people work out their salvation. Maitreya is presently a bodhisattva, but the most important bodhisattva is probably Avalokiteshvara, who became identified with the Chinese goddess of Mercy, Kwan-Yin (Kannon in Japan).
    2. The Buddha was not unique, and individual practitioners who have become bodhisattvas can become Buddhas. There are already multiple Buddhas besides Shakyamuni. Most important are Mahâvairocana and Amitâbha. Amitâbha is famous for his Western Paradise, or Pure Land, where he has Vowed to cause anyone who calls on him for help to be born, so they will be free of the world of suffering to work out their ultimate liberation. In Japan Amitâbha is known as Amida and Mahâvairocana as Dainichi. Most of the famous Buddha statues in Japan are not Shakyamuni: the great outdoor bronze Buddha at Kamakura is Amida, and the Buddha enshrined in the To:daiji ("Great Eastern") Temple in Nara (the largest wooden building in the world), is another Buddha named Locana.
    3. Nirvân[.]a and samsâra are no longer definitely different. The "Fourfold Negation" is applied to the relationship between the two. Samsâra and nirvân[.]a are thus neither the same, nor different, nor both the same and different, nor neither the same nor different. This allows some room for maneuver, which may have made Buddhism more palatable in China, where Confucianism never did approve either of the world-denying metaphysics or the monasticism of Buddhism. Distinctively Chinese schools of Buddhism developed, like T'ien-t'ai (Tendai in Japan) and Ch'an (Seon [Son] in Korea, Thien in Vietnam, Zen in Japan), for whom samsâra and nirvân[.]a were virtually identical, so that enlightenment and nirvân[.]a transformed the world rather than eliminated it. The paradoxical metaphysics of Buddhism could be assimilated to the similar paradoxical doctrines of the native Chinese philosophical school of Taoism.
    
    
    
  • Places where Mahâyâna spread: Mahâyâna Buddhism is presently practiced in China, Korea, Vietnam, and Japan. Buddhism was propagated in China by missionaries from India, like Kumârajîva (344-413), who arrived in China in 401, and Buddhabhadra (359-429), who arrived in 408, and by Chinese pilgrims who traveled to India, like Fa-Hsien (Fa³xia³n), who travelled to India between 399 and 414, and Hsüan-tsang (Xuánza³ng, 600-664), who went to India between 629 and 645. Kumârajâva, Fa-Hsien, and Hsüan-tsang all brought Buddhist texts from India to China and translated them. The Buddhist canon as it arrived in China was in Sanskrit, and it included many special Mahâyâna Sâtras that are not in the Pali Canon (though many are now suspected of being Chinese forgeries). The stories of Fa-Hsien and Hsüan-tsang travels are important parts of Chinese literature, and Fa-Hsien's account of India during the reign of Chandra Gupta II (375-415) is an important document for the history of India.
  
  
  
• Vajrayâna ("Thunderbolt Vehicle") Buddhism: In India, 6th to 11th century.
  
  
  • Distinctive doctrines: Vajrayâna Buddhism is Tantric Buddhism, often called "esoteric" Buddhism. Although it is sometimes rather oddly translated as "diamond," the vajra (ko:ngo: in Japanese) was originally the thunderbolt of Indra; and in Vajrayâna it symbolizes the magical power of Tantrism. Tantric magic could be worked through man[.]d[.]alas, sacred diagrams, mantras, sacred formulas for recitation (the most famous one being, "Om, mane padme hum"--"The jewel is in the lotus"), and mudrâs, sacred gestures. This Tantric magic could be merely thaumaturgical ("wonder working") or could be regarded as means of achieving liberation in addition to or apart from meditative or meritorious practices. Just as Hindu Tantrism expresses its magical power through goddesses like Kâlî, Vajrayâna emphasizes female figures. Vajrayâna comes to balance male Bodhisattvas with female Bodhisattvas as attendants of the various Buddhas. And while Buddhas tend to be regarded as male in all branches of Buddhism, Tibetan Buddhism supplies female figures corresponding to each Buddha, like the "savioresses" Green Târâ, White Târâ, and Mâmakî, who actually vow to always be reborn as women in the process of leading all beings to salvation. Vajrayâna symbolism always balances male and female: the Vajra Man[.]d[.]ala (or the "jewel" above) corresponds to the Womb (or Matrix) Man[.]d[.]ala (the "lotus"). The extent to which Vajrayâna practiced real sexual union as part of its Tantrism is unclear. Often "right-handed" Tantrism is distinguished from "left-handed" Tantrism, in which the former practiced the union of male and female, vajra and womb, in symbolic, iconographic form, while the latter practiced it literally. While the "right-handed" forms are mainly what remain in Tibet and in Japanese Shingon today, there is little doubt that real "left-handed" practices existed in the past; and Tibetan art sometimes still portrays the more violent and disturbing aspects of Tantric practice--rape, bestiality, etc.
    
    
  • Places where Vajrayâna spread: Vajrayâna Buddhism most importantly spread to Tibet and then Mongolia. In Tibet it assumed distinctive forms that are usually called Lamaism, since the monks are called Lamas. The present Dalai Lama, who was the priestly ruler of Tibet until he fled the Communist Chinese in 1959, is from a line that is reputed to be successive incarnations of the Bodhisattva Avalokiteshvara. Vajrayâna Buddhism also entered China, Japan, etc. as special "esoteric" schools, like the Japanese Shingon school. The great temple at Borobudur, about 250 miles east of Jakarta near Yogyakarta on the Indonesian island of Java, dates from this period (c.800), and embodies Vajrayâna man[.]d[.]ala forms; but in Indonesia Buddhism soon thereafter gave way to Islâm.
  
  
  
• The end of Buddhism in India. Buddhism may have died out in India in the 11th century because: 1) It had become almost indistinguishable from the Tantric forms of Hinduism. Sophisticated Buddhist doctrine did not appeal to most people, and the actual practices and iconography of Vajrayâna could easily be assimilated into Hinduism. And, 2) Islâm arrived in earnest in India with the Afghan prince Mah[.]mûd of Ghazna, who defeated a coalition of Hindu princes in 1008 and soon annexed the Punjâb. As conversions to Islâm increased, Buddhism declined. By the time the British arrived, about 25% of India was Moslem. That ultimately led to the partition of the country into India and Pakistan. The Gautama Buddha himself has ended up being regarded as the 9th Incarnation (Avatar) of the great Hindu God Vis[.]n[.]u, although the unflattering take on it is that he deliberately taught a false doctrine (i.e. Buddhism) in order to deceive and destroy demons.