illustration (attribution, if any possible, is at the end of the article)
The Buddha did not describe 'reincarnation', he described 'rebirth'.
When using Buddhist vocabulary, 'reincarnation' is self-based: what's reborn is the same 'individual' (be it thanks to a 'soul' or whatever 'unique' karmic continuity… anything that would make an 'identity' be unique and trackable as the 'same' through change).
Reincarnation is explicitly rejected by the Buddha (according to the Pāḷi Canon and to Mahāyāna sūtras) and by Buddhism in general.
There is a couple of big traps though:
• you can indeed find schools which evolved from Buddhism and talked of reincarnation, but they are not considered Buddhists after such an evolution.
The first people who tried to formalise a misconception of rebirth as reincarnation were the Pudgalavadins and, as a result, they were rejected by all other Buddhist schools (they still imagined and called themselves as 'buddhists' though…).
Many other schools followed a similar path, e.g. when Buddhism was introduced in China: to favour the acceptance of rebirth and karma by Chinese people, many teachers apparently talked about the "indestructibility of the soul" (i.e. the shen-ling doctrine). These misrepresentations were later corrected by most but, sure enough, some perpetuated.
The perpetuation of errors doesn't make them what the Buddha taught, merely the perpetuation of ignorance by people unable to conceive the teachings without an essential 'self' underpinning some of them…
• you can also find teachings for beginners (notably in the Tibetan traditions) which talk of reincarnation, as an entry point into rebirth… but that's like introducing Newtonian gravitation prior to amending it to General Relativity: confusing a simplified, easier-to-follow, introduction with the core advanced teaching is just confusion. (Having different levels of teachings is notably formalised in terms of the "three types of persons" in the Tibetan traditions.)
Self-less-ness is a core teaching of Buddhism, so much so that it's one of the "three/four characteristics", i.e. the three/four criteria used to assess whether a teaching is 'buddhist' or not!
It is because Pudgalavadin doctrines and shen-ling breach self-less-ness that they're not considered Buddhists. You cannot call Buddhist whatever you feel like calling buddhist, that's just abusing the label thus emptying it of meaning. This has nothing to do with a particular teacher picking which schools are 'right' and which schools are 'wrong': there are criteria, shared by all schools, as to what the "characteristics of existence" are (according to Buddhism).
The three characteristics are: unsatisfactoriness, impermanence, and selflessness. A fourth is sometimes added, about Nirvāṇa.
'Rebirth' is not self-based.
Rebirth is about the continuity of tendencies, it is not about 'individuals' and not about 'identity': a legacy of ideas is as much rebirth as the recycling of your atoms in the cycle of life…
One of the interpretations of rebirth goes through what Buddhism calls "dependent origination". This doctrine —once understood, and it's more complicated than it initially seems— makes clear that, often, what's perceived as 'reincarnation' is a mis-interpretation of rebirth: a tendency is appropriated as being yours, but Buddhism explicitly states that this appropriation is a sign of "ignorance" (i.e. not reality)!
The tendency is not intrinsically yours, it doesn't come from your past in any way; you're the one presently making it yours by "ignorantly" identifying to it, you're the one presently interpreting a narrative about the past as being your previous life.
This appropriation has causal consequences: you'll behave differently, anticipate differently, and suffer differently, because of this belief in "who you are". Although it is a mere narrative empty of essence, it thus has 'effects', both psychological and physical (due to the resulting changes in your physical actions, due to psychological factors).
Buddhism explicitly distinguishes 'rebirth' from 'reincarnation', and rejects reincarnation.
Without enough education in Buddhist vocabulary, one of the best equivalent you can use to imagine what 'rebirth' entails would be 'causality'.
While one can observe and characterise causality, one would be hard-pressed to find any "underlying component", or a 'cause' for causality itself. These difficulties don't invalidate 'causality' any more, or less, than they invalidate 'rebirth'. As long as one confuses 'rebirth' with 'reincarnation', these teachings are problematic; but it's not because of 'rebirth', it's because of the confusion with what 'rebirth' is not!
photo © Paulus Rusyanto.