How do pulse oximeters work? does their light have to go through a fingernail or will any part of the finger do?

In fact,: the oximeter is so sophisticated that it only measures the substance that has "changing absorbance" (arterial blood"); it is able to disregard "non-changing absorbance" i.e. tissues like skin, muscle or nail. The fingernail, then, has nothing to do with the measurement. Hemoglobin in blood absorbs different light wavelengths; these are compared and computed to come up with the oxygen sat. Amazing!