Generally in the West karma and reincarnation are not accepted because neither Christianity nor Judaism include it. But modern spiritually inclined people are usually oriented towards Eastern religions, which include karma and reincarnation. I have to differentiate here between Western spiritually inclined people in general and specifically Western Advaita seekers. While the latter are not necessarily concerned with it because the law of karma refers to duality, others are fascinated by it. I will call them here “New Age scene”. They have their very own theory of Karma (which occasionally is shared by Western Advaitins as well). Although the terms reincarnation or karma may be used all the same, different ideas go along with them.
First I’d like to outline the concept of Advaita Vedanta at least in part.
While the law of karma does not present absolute truth in Advaita Vedanta, it is considered as one of the most profound laws in transactional reality. The law of karma helps the seeker to see life in a bigger context, take responsibility for the fate that befalls him or her and surrender personal desires to the universal law and order (Ishvara).
The law of karma is the expression of the law of cause and effect, which we constantly apply in our daily life:
1. Anything you do (karma = action) will produce an effect.
2. Anything that occurs in your life is either due to an action of yours in this life or an action of yours in past lives.
3. Any good action will produce something that will affect you positively; any bad action will produce something that will affect you negatively.
4. The results of actions will come about at different speed, some needing hours, some many lifetimes.
5. The total of the potential results of your actions collected over lifetimes is called sanchitta karmas.
6. Out of this total the portion that is due at the time of your birth is called prarabdha karma. The prarabdha karma is wedded to the body-mind-system of a certain lifetime.
7. So with reference to 2 above: anything that occurs in your life is either due to an action of yours in this life or due to your prarabdha karma.
8. Anyone who has not yet realized truth, will inevitably create new karmas in his lifetime; the potential results of those are summed up under the name agami karma. The agami karma will be added to the sanchitta karma following the death of the body.
9. The resolution or exhaustion of prarabdha karma will result in the immediate death of the body.
While Western seekers usually have some vague idea about reincarnation and karma, it is by no means as detailed as the above nine points, which form just a portion of the whole picture. Still, most of those being associated with the New Age scene have adopted the idea of reincarnation.
As they belong to quite a secular culture, they usually dismiss the idea of God. So they assume that they themselves have chosen their particular body-mind and life-circumstances. This leads them to assume responsibility for whatever does or does not happen in their life and in that way it could have a relaxing effect. But this relaxation is usually lost because the Western mind seems to be always out to get somewhere; it is always future oriented. To be specific, connected with the idea of responsibility is another one, also foreign to the original concept of karma: typically karma is seen as another word for assignment.
The whole theory goes like this:
1. Karma is some residue of past lives, signifying something that I did not quite understand, did not live out, a mistake that I committed or something bad that I did.
2. Because this did not get resolved in the past, I am presented with the issue again in order to get another chance to resolve it.
3. As soon as I understand the point, lived out whatever I avoided, corrected the mistake or bad deed in a similar context in this life, the issue/ assignment/ karma is resolved.
4. Good karma is not taken into consideration because if I did everything fine, I need not learn anything, so no need for an assignment (= karma).
5. The death of the body is unrelated to karma. Death can happen any time.
So in the West the whole theory of karma remains psychological; it is still about personality development. Not surprisingly these New Age seekers usually have no interest in ending the cycle of birth and death because there still remains so much to do, to experience, to learn, to grow, to refine and to break free from.
In traditional Advaita Vedanta the law of karma has nothing to do with assignments. Learning is most welcome for every human being; in the end personal development will help the person move from being action-oriented to being knowledge-oriented, enabling him/her to be interested in Vedanta and have the chance to realize his true nature. But learning and personal development is considered to be a mere side effect of the law of Karma.
Karma is resolved as soon as one has experienced the result of a past action – no matter whether I learned something, whether I lived something out, whether I corrected my mistake or bad deed. If I did, fine, possibly I will produce better agami karma in this life but the point is that I had to experience the result.
While the law of karma for the New Age scene represents a method of self-improvement, for Advaita Vedanta it is a means to relax the mind in order to open it up to know that which is beyond any karma.
 For further information please refer to an article posted by Kuntimaddi Sadananda on this site: http://advaita-academy.org/talks/Reincarnation.ashx