Buddhism and all its major versions

The core version

Souls exist (but if you look deeper, not really), and they reincarnate repeatedly into many different forms of creatures, such as gods, humans, non-human animals, plants (maybe), hungry ghosts, and hell-beings. This cycle of rebirth is the samsara.

The world is extremely big, time is cyclic, and there are many universes. (These ideas are taken from Hinduism.)

Reincarnation is determined by a balance of karma. Karma is added by good actions and suffering bad events, and vice versa. This is why living in comfort loses karma, and having a bad life (such as those in hell) gains karma.

Life (except the gods') is full of suffering. There are four great life events, and every one creates great suffering: birth, old age, sickness, and death. Further, the suffering doesn't stop at just one life, because souls are reincarnated endlessly. Even the gods are not immune because gods eventually die too and they can get reincarnated to a suffering life.

Existence is fundamentally empty, in the sense that you do not have a fundamental nature, no self, no soul (anatta). Rebirth is accomplished by the same trick that makes you think you are the same person you were yesterday, even when you aren't. In short, personal identity is a stubbornly persistent illusion. I think Feynman stated the same idea here, musing on the fact that our body do not keep the same atoms:

To note that the thing I call my individuality is only a pattern or dance, that is what it means when one discovers how long it takes for the atoms of the brain to be replaced by other atoms. The atoms come into my brain, dance a dance, and then go out—there are always new atoms, but always doing the same dance, remembering what the dance was yesterday.

The final solution is to achieve nirvana, which is freedom from life, time, and the cycle of rebirth, by becoming absolutely nonexistent. The word "nirvana" literally means "blown out", and a person achieving nirvana is like an oil lamp blown out. They aren't gone to a heaven, but completely nonexistent. No thought, no nature, no self, absolutely nothing.

To achieve nirvana, one should perform certain actions and learn certain things. The popular ones include: meditation, study and recitation of sutras, be vegetarian, do charity work, etc,

Brief history of the spread and fragmentation

Buddhism was started by Gautama Buddha in 5th century BC in a part of India, before spreading out to India, China, southeast Asia and Japan, etc. See the map below.

In modern times, there are three major schools of Buddhism, as mapped below:

Theravada: The oldest school

This is the conservative version, and out of all the major schools of Buddhism described in the post, this is the one closest to the original Buddhism that Gautama Buddha taught. It's close enough to the "core version" I described above that I won't go into the details.

Mahayana: The Chinese and Japanese version

Buddhism became popular in China around the 400s, and it developed into Mahayana. It spread out of China into Japan.

Mahayana's main difference from Theravada is that it emphasizes on saving all sentient beings, instead of only saving yourself. Mahayana Buddhism denounces Theravada Buddhism's goal of becoming an arhat (an enlightened being that has escaped samsara), and requires anyone who has become enlightened to not disappear into nirvana, but rather engage with sentient beings that are still stuck in the samsara and help them escape as well.

Because of the engagement with the world, Mahayana has a much bigger pantheon, with many Buddhas and Bodhisattvas.

A Buddha is someone that has achieved nirvana and thus out of this world, but still somehow connected to this world. This problem is the same problem that haunts Christianity, about how God could be totally independent . Each universe usually has at most one Buddha.

Bodhisattvas are basically lesser Buddhas.

It has two main branches, pure land, and zen.

Pure land Buddhism

Pure land Buddhism adds an extra type of universe, called pure lands. Those are special universes created by some Buddhas. These pure lands are beautiful, nice, and a great place to continue the spiritual path towards becoming a Buddha.

The most important pure land for our universe is made by Amitābha, and it is to the west of our universe (I don't know how "west" works in the multiverse, though).

Pure land Buddhism concentrates on getting to the pure land as the main intermediate goal for the practitioners, with the idea that, after reaching pure lands, Buddhahood is much easier.

Chan/Zen Buddhism: the illogical

Chan Buddhism developed in China around the 800s. Its distinguishing feature is that it uses a lot of weird and illogical ways to achieve enlightenment. The zen koans are the most famous examples of its weird ways.

• What is the sound of one hand clapping?
• What is your true face before your parents were born?
• What is this corpse walking about?

The main point of these zen koans (as well as other weird methods) is basically to cause enlightenment by breaking down certain strongly-held illusions of duality, which Chan Buddhism claims cannot be achieved by rational reasoning, but by strategic confusion and shock. Illusions of duality are like gender: they aren't real, but are seen commonly enough that one imagines them to be a part of fundamental nature. Like the duality between determinism and freedom, between life and death, self and other, human and nonhuman.

Vajrayana: the esoteric and weird

The main feature of Vajrayana is its esoteric way of teaching. 

It is esoteric, meaning that its teachings are done not by copyable texts, but by the personal guidance of a teacher, usually kept secret from outsiders. This makes the study of Vajrayana difficult for scholars.

Its main practices are called the tantras, which all seem to be kind of weird, irrational and scary. There's the mandala, which is staring intensely at a colorful complex picture and meditate on that. There's the visualization technique, which is just imagining yourself as a god and somehow it would help you become a god. There are many, many rituals, which are just Buddhist magic. There are "dark arts", which is the strategic use of the "bad" parts of life, such as sex, blood, meat, drugs, ritual instruments made of bones. 

Out of all the tantras, the sexual tantra seems to be the only thing that got into Western pop culture.