Reincarnation and karma are core concepts for several religions of India, including Hinduism, Buddhism, Jainism and Sikhism.
Basically, both concepts have to do with an understanding of time and how we as human beings are propelled forward through life in time.
Reincarnation is sometimes called transmigration of the soul. To believe in this is to believe that the soul migrates through many different physical lifetimes. The soul undergoes rebirths into different kinds of lives - human, animal or even supernatural - until it reaches its final destination.
So, reincarnation implies a cyclical understanding of time instead of a linear understanding. In the Western world, people tend to think of time in a linear way - you are born, you live, then you die. Depending on one's belief about afterlife, the soul may live forever in a place of suffering or blessing. But, it does not return to earth to live again as a human or animal.
Those who believe in reincarnation, however, hold to a cyclical view of time in which the soul undergoes cycles of life, death and rebirth. The soul may be born into a human body and then, when the human body dies, it may be reborn into an animal body or the body of a supernatural being (angels, demons, etc.). This cycle may be repeated many hundreds of times.
What propels the soul forward from life to life? What determines the kind of rebirth or life the soul will have in the future? The answer is karma.
Karma refers to the energy that accompanies actions of moral worth. Not all actions have moral worth. For example, you may pick up a stick. In itself, that action has no moral worth. However, picking up a stick and hitting someone with it does have moral worth - of the negative kind. On the other hand, picking up a stick and using it to plug a hole in a levee so that the city is not flooded also has moral worth - of the positive kind.
Karma is the positive or negative energy that accompanies all actions of moral worth. Hitting someone with a stick brings negative karma to the person who does it. Using the stick for a positive reason brings positive karma to the person.
So, each person - each soul or atman - goes through life doing both positive and negative actions (hopefully more positive than negative) and accumulating the karmic energy or "baggage" that comes with them. The nature of one's karma determines the nature of one's rebirths in futures lives. Negative karma over many lifetimes will cause an unfavorable rebirth as an animal. Positive karma will cause a favorable rebirth as a human or auspicious spiritual being.
This belief in karma and its impact on future rebirths is at the heart of basic ethical sensibilities in Hinduism and in all the other religions that affirm it. People should do good actions and avoid bad actions so as to not accumulate bad karma to themselves.