NK cells patch a hole in the body's defenses. All nucleated cells display a cluster of molecules on their external surfaces called Major Histocompatability Complex Class I (MHC I), and in this they place a sample of the proteins being made within the cell. Why do they go to the trouble? Principally to foil viruses and tumours, as we'll see.
Cytotoxic T lymphocytes are a lethal subdivision of the body's police force. If provoked, they release their granules and thereby either cause the cell to auto-destruct (apoptosis) or else, failing that, simply kill the cell themselves. (They release granules which punch holes in the cell, amongst other things.) The way they identify a virally-infected cell (or a tumour cell) is by checking the proteins presented to it on the cell's MHC I molecules. An innocent cell would simply be presenting normal protein fragments as usual, but a cell that is under the spell of a virus would unavoidably end up displaying some part of the nascent virus on its MHC I molecule. If a cytotoxic T lyphocyte should notice this, it immediately sets about obliterating the cell, thereby combating the virus' ability to spread. (A similar picture would emerge for tumour cells, which also produce abnormal proteins intracellularly.)
Hmm. If you were a virus (or a tumour cell), what could you do to avoid this fate? Why, simply make the cell produce as little MHC I molecules as possible, surely! That way, what goes on in the cell stays in the cell, so to speak, and the cell would be relatively insulated from cytotoxic T lymphocyte attack. This is exactly what certain viruses have done - the herpes viruses in particular.
This is the hole in the immune system's defenses that I alluded to, and it is simultaneously the raison d'être for natural killer cells. What they do is scour the blood stream looking for cells that display low levels of MHC I molecules. NK cells are covered with receptors for MHC molecules and when they bump into a normal cell its MHC molecules bind to the NK's receptors, thereby sending an inhibitory "don't kill me!" signal. Should any cell not have enough MHC I molecules, the NK cell activates itself and kills the target in much the same way as the cytotoxic T cells do. It's really a system of such elegant simplicity.
[For completeness' sake, NK cells can also kill cells that are coated with antibodies, but this ability was probably a later addition to their original job description. This phenomenon is called antibody-dependent cell-mediated cytotoxicity (ADCC).]