Buddhism does not use the word reincarnation, it uses the word rebirth, because in general in English we think of the word reincarnation as a permanently unchanging self that's being reincarnated. Reincarnation is a type of rebirth also, but my understanding is that it is something solid that gets reborn over and over.
With the definition of rebirth in Buddhism it's not a permanent unchanging self, but more a moment to moment process of cause and effect that is reborn so we use a different word.
The conception, when your father and mother got together and the sperm went into the egg, at that moment Buddhism says consciousness came, you started right there. It was a rebirth only from the last consciousness when you died. It was not a rebirth of something that's been living forever and ever, but only a rebirth from the last conscious moment, of your Kamma makeup wherever you were when you died, that is reborn. So this is different to reincarnation that more or less states there's an essence that's always there. Buddhism says no, we're changing all the time. And we're being reborn each moment. You walked in here, a certain type of person, with a certain type of understanding - you now have 40 minutes more of understanding. You've been reborn with a different type of understanding in your mind. So that's what we mean with rebirth, it's a moment-to-moment change.