Vinnana. The Pali word for what gets reborn is Vinnana. Consciousness is the best English word we can use for it. We can understand it in this way: Some of you are looking directly at me, so your eyes are looking at a sight and your mind knows what you are looking at. Some of you are only listening. Your ears are listening to sounds but your mind knows what you are hearing. The mind's knowing, that quality of knowing, is Vinnana. Consciousness, an awareness of what is.
Vinnana, to a certain extent, gets also interchanged yet confused with another Pali word, Citta. Citta, in general English, is referred to as the "heart-mind" and the Thais point here [center of chest] when they point to the Citta, their mind-heart. English has no equivalent of this concept. English talks about soul, talks about a consciousness that is a bigger thing, etc. These are all a lot of words, often difficult to actually explain, and to a certain extent it doesn't have to be explained.
This is one point where a lot of words can send a person round in a circle. What gets reborn? Buddhism will say Vinnana, this consciousness, is reborn, but basically it is connected to all of your Kamma. So in a sense your Kamma gets reborn. That gets a little more wordy and hard to understand, but the consciousness that you die with latches onto another body, bringing with it all its Kammic make-up. So if you were a loving person before then you will generally have a loving new rebirth wherever you are, and as I mentioned before, loving produces beauty and so on, things like that. So the consciousness moves to another body.
We could also say that the "mind" gets reborn. But again these are all just different words, which I hesitate to get involved with too much. We have to understand that it is not a solid entity that is always getting reborn from body to body to body, and that the person you are when you die is going to be different to the person you were when you are born, that the consciousness is different. The consciousness changes, it is an always changing thing.
Now, a caution on this whole area, we've talked a lot about Buddhist theories in the last few days. A caution is that it is ok to understand about the theory of Buddhism, but not to get so caught up in it that you really want to work it out, and figure out all the details and read everything about the theory of Buddhism. Because many people get lost thinking about the practice rather than doing the practice. We can actually see the rebirth concept in a moment-to-moment reality here now. As I mentioned before, if we are loving, people like to be around us, they think we are beautiful, because we are loving in that moment. If we are angry, people don't want to be around us, they feel that we are ugly in that moment. So we can see this in a practice level and that is really the most important aspect, we've got to get down to the practice level.
Theories can help us see a bit of the framework around Buddhism, and often we then see that theory actually works, and that is helpful. Also the theory is beneficial because it counterbalances the theories that we grew up with in our Judean or Christian societies, heaven and hell, etc. This helps to balance being attached to either one of the theories, and to be more open.