Buddhism talks about the Five Aggregates. If you remember in Rosemary's Satipattana Sutta talk, the Buddha taught that we are made up of five aggregates or parts. The body is one aggregate, and the mind processes are divided into four aggregates. As I understand, our Kamma element, all that we have done, all that we have said and thought, all our past actions , speech and thoughts, is tied in here but is not specifically one of the aggregates. But supposedly the aggregate of consciousness, Vinnana, together with our past Kamma that propels us to the next birth.
In Thai Buddhist teachings there are two different Pali words that get translated into English using "consciousness."
One is the word "Vinnana" which is one of the five aggregates and it is a process of the mind that "knows" things.
Then there is the word "Citta," for which the Thais point to the heart, calling it a "mind-heart," also sometimes called "consciousness." It is a very difficult kind of word to use because consciousness in the West also often implies some essence, some soul that gets reborn. Re-incarnated is another word that gets used elsewhere.
It is taught that, what you inherit - your body, your parents, benefits or non-benefits, is due to your Kamma. It is all due to your past Kamma. So, when you think about what is getting reborn, if you make your Kamma really good, Buddhism says you will get a good rebirth. If you don't make your Kamma really good, well...