What does reincarnation mean to Buddhist?

There is more than one understanding of what rebirth is according to Buddhism.

In general, a traditional view is that if one is not enlightened in this life, then one still generates "the fruits of one's actions" (the results of karmic actions) and the remaining fruits that have not been applied to this life will be applied in the next life. What exactly gets reborn, in this cosmology, is still debated.

Modern western Buddhism has developed the understanding that when the Buddha spoke about rebirth he was using it as a metaphor -- that what is born and reborn is our created "self" -- the one that gets mistaken for our eternal selves, our "atman", our "soul". It is our created self that keeps us stuck in a cycle of delusion, greed, ill will, ignorance -- a literally "self perpetuating" cycle. When we see this accurately we can practice skills that will help us break out of it.