What is reincarnated? What is reincarnated? Does reincarnation occur in the realm of conventional reality?

Well I had to put in some superfluous words because the website wouldn't accept just a simple question.
It appears original Buddhism did not teach about "reincarnation". It appears "reincarnation" was introduced into Buddhism at a later time. 

The Pali suttas show original Buddhism taught about "upapajjati" ("re-arising"), which means a certain result arising following from certain actions. "Upapajjati" is concerned with the results of kamma or morality. 

The original Pali suttas most often say what is "re-arises" is the former sense of "being" or "self".  

For example, you have the idea: "I will rob this bank because I want to have money". Then when you are imprisoned for robbing the bank, you have the painful regret: "Why was I so stupid to get get caught?". What "re-arises" is the same sense of "being", "self" or "I"; that is caught up or strongly attached to the idea of robbing a bank for money. 

The scriptures say: 


  'These beings — who were endowed with bad conduct of body, speech &amp;
  mind, who reviled noble ones, held wrong views and undertook actions
  under the influence of wrong views — with the break-up of the body,
  after death, have re-appeared in the plane of deprivation, the bad
  destination, the lower realms, in hell. But these beings — who were
  endowed with good conduct of body, speech &amp; mind, who did not revile
  noble ones, who held right views and undertook actions under the
  influence of right views — with the break-up of the body, after death,
  have re-appeared in the good destinations, in the heavenly world.'


Therefore what "re-arises" is "a being". The scriptures also say "a being" is a mental state of strong attachment, as follows:


  'A being,' lord. 'A being,' it's said. To what extent is one said to
  be 'a being'?"
  
  "Any desire, passion, delight, or craving for form, Radha: when one is
  caught up there, tied up there, one is said to be 'a being.'
  
  "Any desire, passion, delight, or craving for feeling... perception...
  fabrications...
  
  "Any desire, passion, delight, or craving for consciousness, Radha:
  when one is caught up there, tied up there, one is said to be 'a
  being.'
  
  SN 23.2


While "a being" is a mental state of delusion; it is this delusion that "re-arises" or "upapajjati".