Pulse oximetry exploits a physics phenomenon: oxygenated haemoglobin absorbs different wavelengths of light than does deoxygenated haemoglobin. Actually this phenomenon is really common - the reason that leaves are green is that they reflect light in the 'green' wavelength back to us whilst absorbing most the rest of the light. Bricks do the same, except that for them they absorb most of the 'not-red' wavelengths of visible light, reflecting the remainer and thus rendering them red.
So one would almost expect that deoxygenated haemoglobin absorbs different wavelengths than oxygenated haemoglobin does (technically, it has a different 'absorption spectrum'). And it does. A pulse oximeter fires two wavelengths through our skin - red and infrared, which are each absorbed preferentially by one of the two species of haemoglobin under consideration. When these wavelengths travel through our capillaries, they are differentially absorbed in a ratio that corresponds to the ratio of oxygenated to deoxygenated haemoglobin. A computer does the calculations, and gives you the arterial saturation of the blood.
Hope I got the physics right! Clever, isn't it?