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Disclaimer: My answer to this will be non-exhaustive. My answer will be scoped as close to that of the question as possible. (General). I will also answer your question (which is actually 4) piece by piece.
You asked:
Does reincarnation/rebirth occur in Buddhism?
Yes it does. The concept of rebirth and reincarnation in Buddhism is complex and can differ slightly or largely from tradition to tradition however.
You asked:
Will everyone be reborn?
Every sentient being (people, animals, etc) is in a cycle of rebirth. They will all be reborn after death. The only exclusion to this will depend on the tradition. Most traditions, in a general sense, state that one may leave the cycle of rebirth if they attain enlightenment or nirvana. This is only possible when you are a human, though.
You asked:
Are people always reborn as people?
The answer is no. Please see next section of this answer for elaboration.
You asked:
How is rebirth/reincarnation handled in Buddhism?
In Buddhism, there exists the concept of Samsara. This is the continuous cycle of rebirth throughout six different realms. These realms include the realm of Gods, Titans, Hungry Ghosts, Hell, Animals, and Humans. One may be considered more preferable to another, for example Gods to Hell, yet all are imperfect and involve some level of suffering. Depending on past lives and karma, one will move from realm to realm during each rebirth. The Human realm is generally considered to be the best because it is the only one in which one may obtain enlightenment and exit from this wheel of Samsara and therefore end the cycle of rebirth and suffering.
For more information, you may find this article to be useful.
Side Note: My professor of Eastern Religions once told me that being in this cycle of rebirth can be a great source of anxiety, hence one source of motivation to exit from it.
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Simple answers for last three questions as far as I know,
Does it occur ? Yes it is
Will everyone ? Yes until someone attain Nibbana
always as people ? No, it can be any(animals/human etc)
But the first question is too broad to answer.
You can find more details in this article about recantation.
Simple explanation for recantation as according to that article,
Reincarnation is not a simple physical birth of a person; for instance, John being reborn as a cat in the next life. In this case John possesses an immortal soul which transforms to the form of a cat after his death. This cycle is repeated over and over again. Or if he is lucky, he will be reborn as a human being.
Upvote:2
How is the concept of rebirth or reincarnation handled in Buddhism? Does it occur? Will everyone be reborn? Are people always reborn as people?
Strictly speaking there is no reincarnation in Buddhism, but rebirth is. 'Reincarnation' imply idea that some constant Soul incarnates into being, which 'rebirth' does not imply. Moreover, Buddhist teaching deny existence of such constant Soul. After death, new being is born, who causally related to past being and carry on actions of past births. Everyone, except who attained final Nirvana, rebirths in almost any place in the Cosmos depending on his/her past actions (karma).