What, in a nutshell, is the law of kamma?

The Buddha said that the essence of kamma is intention. The law of kamma (or in Sanskrit: karma) expresses the moral dimension of the law of cause and effect. Any intentional action performed through body, speech, or mind produces results consistent with that intention. Put most simply: good actions have good results; bad actions have bad results. Actions provoked by toxic mental states rooted in greed, hatred and delusion contribute to future suffering. Actions flowing from wisdom and compassion contribute to future happiness.

Is everything that happens in our life meant to be, or is there such a thing as free will?

The Buddha rejected the belief that everything in our life is fated, preordained by a supernatural power. He also encouraged his disciples to see how the idea of an independent self-exercising free will disappears with close analysis of the body and mind.

At each moment of our life, experience has a certain tone: pleasant, un-pleasant or neutral. Lacking mindfulness and wisdom, we react to the pleasant with grasping, the unpleasant with rejection and the neutral with dullness. In this way, our life is largely determines by habitual reactions to the raw material of experience. With mindfulness and wisdom, we recognize the affective tone of experience as just that, but make decisions based upon more intelligent criteria. In this way, a certain freedom from the given may be known.

Please illustrate the workings of the law of kamma

Every day we perform so many volitional acts. Our life is such a complex flow of volition, that the effect of any one particular act is rarely obvious. To use an analogy, is a bucket of acid were to be thrown into a river, we could be sure that it would reduce the pH level of the water to some degree. But the extent to which that change is observable would depend on what other substances had been introduced into the water. Is water was already very acidic or very alkaline; the effect might not be at all obvious.

Although the external effects of individual kammic actions may not be easily verified, internally it is a different story. We can easily observe that every time we indulge in anger, we increase the likelihood that we will indulge in the same way in the future. We create and feed habits and personality traits through a constant drip of volitional actions. Every time we act with a coarse intention, we immediately become a slightly coarser human being. Every time we act with kindness we immediately become a slightly kinder person.