This Sanskrit word means 'action.' The concept of karma is near-ubiquitous in esoteric discourse of different sorts and origins and many shades of interpretation exist. Generally, the concept involves a soul's 'accountability' for its actions and is usually understood to cover multiple lifetimes. One eventually reaps what one sows is the general idea.
From the 4th Way viewpoint, it only makes sense to speak of karma if man has evolved beyond being a chance combination of influences, possesses an I that survives physical death while maintaining some recognizable cohesion, enjoys free will and is not simply a reaction machine. Because the 4th Way teaches that man in the overwhelming majority of cases does not meet these criteria, it does not particularly emphasize any teaching about karma, but rather concentrates on making students such that the concept might one day apply to them in a true sense.
Mouravieff writes about a principle of recurrence governing lives of the exterior humanity. This involves a sort of karmic carry-over between rounds but is not really reincarnation in the sense of one consciously choosing incarnations for a definite aim. See Recurrence. Ouspensky also speculates about recurrence. Gurdjieff says that something of the sort may exist but that it is quite immaterial to talk further about it until the student be much more advanced than Ouspensky at the time.
A simplistic interpretation of the idea of karma would be that any 'bad deeds' will return to the perpetrator in such a way that the perpetrator will come to renounce all such badness and see the error of one's ways first hand. Thus, over lifetimes, all becomes perfected towards a universal idea of goodness, truth and beauty. Observation of history does not support such a view.
Rather, it seems that the principle of karma allows for two different paths of evolution, corresponding to service to self and service to others. It may be that the eventual price of unbounded service to self is some sort of dissolution but the being gets to develop in power and capacity to far above human levels before anything of the sort takes place. Thus karma does not make a 'fair world' not at least from the human standpoint.
'Doing good' in the hope of a karmic recompense is also a likely misuse of the concept. Following a path of service to others is not service to others if it is done for a selfish goal. 'Things come not to those positively oriented but through such beings.' [Ra]
Much has been spoken of the mechanism which would enforce karma. Most often it is said that a sort of 'higher self' makes the determination on how a life should be balanced. To a degree, the principle of karma could be seen as a tour guide of the 'far country' of incarnation, showing the reincarnating entity different sides of all dynamics. This idea is however simplistic and does not take into account the notion of polarization, which clearly seems to exist. The same Trials may increase one in the strength of commitment to a path of service but may turn another into an entirely opportunistic predator.
Rudolf Steiner has written extensively about karma, at both personal and collective levels. He offers many case studies, including Nietzsche, through which he seeks to explain specific laws which mediate between incarnations. The material is derived from his own clairvoyant faculties and is no more verifiable than channeling but does offer interesting models for consideration. Steiner proposes specific progressions of states, for example, intense hatred in one life becomes pain in the next and lack of intellect in the one thereafter. Love in one would be joy in the next and perceptive ability in the one thereafter.
The specifics vary but most sources accept that as a soul becomes more individuated, i.e. more constant between cycles of recurrence/reincarnation, the entity gets to choose the circumstance and work consciously on karma. Otherwise this is a mechanical process of cause and effect.
If something reincarnates and is subject to karma, then what is this? Nobody seems to have a precise definition. The Buddhists see man as a composite of five 'skandhas' or substances. The Gurdjieffian 'hydrogen' may somewhat correspond to the concept of skandha. . During a lifetime, man may produce various crystallizations and wholly or partially formed higher bodies which may persist after physical death and undergo further experience, including incarnation.
The skandhas are a sort of wave packet in the 'Sea of Samsara,' which roughly means the terrestrial world where incarnation takes place. Some patterns are more cohesive than others and sort of keep cycling between incarnate and disincarnate states. There is no strict permanent identity of individual, although there is preservation of some 'crystallized' traits and gradual morphing, amplification or attenuation of some 'frequencies' in the 'wave packet of skandhas.' Karma might be compared to the laws of physics governing the propagation of these waves between incarnations. Sometimes, a well-enough formed wave packet can have a measure of 'free will' and utilize the laws of physics or karma towards one or another direction of development.
While the terms and cultural tone are different, we see some parallels between the Buddhist idea of identity and the Gurdjieffian one.