You have post a great topic that I am really interested in.
What I have learn from the fourth noble truth, rebirth and karma (covered closely the second noble truth) taught Buddha by has noting to do with reincarnation after life. But various schools could have taught about this differently.
Rebirth is the is rebirth of the karma across life. Karma is in this sense is not a cosmic system of punishment and rewards, but the result of the thoughts, feelings and daily action that we take that influence others and triggering the resulting actions of the others. Rebirth of karma across life means that resulting same actions of others reborn in other generations.
Example, a parents push the blame to others on everything and others when a faults occur, thus his kids to grew up and learn to push the blame on others and everything, when his kid grew up, the grandson learns the same thing, and results in the rebirth of the karma of blame, pushing the blame to others on everything. The pushing of blame to others resulted in the rebirth of the same thing through different generations.
Now if the parents is to stop the karma of pushing the blame to others, the kids will not learn the same thing, and in returns the grandson will not learn the same thing. Thus stopping the rebirth of karma.
Rebirth can also happen within self, let's say today you got angry at someone. In 5 years time the same thing occur and you get angry at someone with the same things. That's the rebirth of karma within self.
Think of us in ocean with different people emitting different karma, in this ocean, different or the same people may emit the same karma in the future.
To reach enlightenment means that to put a stop to rebirth and karma, meaning to be as a rock within the ocean not not affected by any karma and rebirth, and not to emit them.
I may be wrong with my explanation, afterall, there would be others like you who is more knowledgeable than me in this. This are just my thoughts regarding rebirth.
Lastly, have a great day ahead.
My Source: ://buddhism.about.com/od/karmaandrebirth/a/karmarebirth.htm