The understanding of Buddhist rebirth and karma is a subtle matter, especially if one is not clear what a Buddhist means by these terms. Buddha Shakyamuni taught in a certain cultural context. This caused him to teach using certain traditional Indian words such as karma.
However, Shakyamuni's use of the words differs from how the Vedic literature used them. In addition to this, in the West we have many many 'new age' ideas as to what karma means. These other meanings are not 'wrong', but they are not the Buddhist presentation of karma.
On the internet are many excellent Buddhist resources, no longer are we limited to the early translations and texts that were the only available works in the 70s. For example,
www.berzinarchives.com is the website of one of H.H.D.L's English translators.
This site has tons of info; on the issue of karma at a basic level and in general terms:
"Question : Is the theory of karma empirical and scientific, or is it accepted on faith?
Answer: The idea of karma makes sense in many ways, but there is some misunderstanding about what karma is. Some people think that karma means fate or predestination. If somebody is hit by a car or loses a lot of money in business, they say, "Well, tough luck, that is their karma." That is not the Buddhist idea of karma. In fact, that is more the idea of God's will - something that we do not understand or have any control over.
In Buddhism, karma refers to impulses. Based on previous actions we have done, impulses arise in us to act in certain ways now. Karma refers to the impulse that comes into someone's mind to invest in a stock the day before it crashes or before it rises in value. Or, someone may have the impulse to cross the street at just the moment when he or she will be hit by a car, not five minutes earlier or five minutes later. The arising of the impulse at just that moment is the result of some previous action or actions the person did. In a previous life, for example, the person might have tortured or killed someone. Such destructive behavior results in the perpetrator experiencing a shortened lifespan as well, usually in another lifetime. Thus, the impulse to cross the street arose at just the moment to be hit by a car.
A person may have the impulse to shout at or hurt someone else. The impulse comes from habits built up by previous similar behavior. Yelling or hurting others builds up a potential, tendency and habit for this type of behavior, so that in the future, we easily do it again. Shouting with anger builds up even more of a potential, tendency and habit to make an angry scene again."
Likewise, Dr. Berzin presents an easy to understand explanation for Buddhist rebirth, which does NOT involve any independent and solid soul or self moving from body to body.
See this site for more basic info on Buddhist karma and rebirth:
http://www.berzinarchives.com/web/e...ry_material/basic_question_karma_rebirth.html
As stated by H.H.D.L., if western science 'disproves' the Buddhist doctrine of rebirth, it should be removed. However, this is not the case as that Dr. Berzin makes clear in the above link.
Buddhism is not a belief system that must be 'accepted' all at once. It is a gradual unfolding of understanding. However, within every traditional Buddhist lineage to exist (including Theravada, Tibetan, Zen, and so on) there is a belief in literal rebirth and karma.
The tricky part is of course understanding what is meant by these terms, and the above links and websites can help in this regard.
Although these are core tenets, different traditions will present them from different angles. Additionally, teachers will teach them differently to different students based on the students capacity and current understanding.