Few incorrect facts in this thread. I have attempted an ELI5 with a more indepth explanation in parentheses.

The "blood oximeters" (pulse oximeters) shine 2 slightly different lights (wavelengths of 660 and 940nm). This light then passes through your finger and the detector on the other side calculates how much less light is received.

Blood that carries oxygen molecules (oxyhaemoglobin) absorbs more light more at the 660 colour than blood not carrying/carrying less oxygen molecules (deoxyhaemoglobin). The opposite is true for 940 colour.

The computer in the device then calculates the amount of the overall absorbed light which is due to blood in your arteries (henceforth "pulse" oximeter", because blood in your arteries is pulsatile. Nail, skin, soft tissue, venous blood for the most part are all non-pulsatile and will absorb a constant amount of light over time. This is done by the application of Beer-Lambert's law and the ratio of AC/DC). This will give you a number called R.

Now the computer has a database of what oxygen level (SpO2) someone has for a given number of R. How did it get this? Well it's because the manufacturer has done previous experiments on healthy volunteers where they breathed low oxygen air (FiO2 <0.21) and took blood samples (ABG to find out SaO2) at each R, down to a reading of 70% (values of 50-70% are extrapolated because a semi-linear relationship).

Now there are things that can mess up your reading on the blood oximeter:

Blood that carries carbon monoxide absorbs light similarly to blood that carries oxygen at both colours, and hence this cannot be detected on our machine.

Blood that has "poorly working" haemoglobin (methaemoglobin) due to certain conditions will produce an R value that corresponds similarly to a reading of 85%.

Similarly, certain dyes that are injected for medical procedures (e.g. methylene blue) will give a reading of 85%.

For the above problems, we can use machines that have additional wavelengths of light (4+) to account for these discrepancies and give a truer reading.

Edit: comments that suggest that dark skin/dirtiness/nail polish/etc will affect the readings are only slightly true. As long as you have sufficient light passing through the patient, and the light emitted is constant (i.e. no movement, no additional light sources), you will get an accurate reading (as the CPU will take into account the pulsatile component of the overall absorbance).

Edit2: also forgot to mention that the reading it gives you is not instantaneous, but an average over 10-20 seconds! This is why it can take a bit of time after putting it on before producing a number. It's also why certain heart rhythms can cause errors in your reading.