For most watch enthusiasts, seeing the prefix of 316L in stainless steel is often enough to say that this is what you should be looking for in a quality steel-cased watch. Without getting too much into metallurgy, 316L is a standard for an austenitic variety of stainless steel containing 2~18% of chromium, molybdenum, and carbon, and less than 1% of other elements like manganese and silicon.
Its main desirable attribute is its resistance to corrosion making it suitable for many applications.
It's called surgical grade because of its use in biomedicine in bone screws, surgical implants and instruments, as well as body piercings and jewellery. Its other useful trait is that it does not cause a reaction with living tissue. 316L however, is nickel-free and is more of a subgrade of 316 (minus the L suffix) which does contain nickel that may trigger a person's immune system or cause an allergic reaction.
316L is not the only grade of steel used in medicine. Cutting instruments can also use other grades like 440 or 420 and are also called surgical grade steel. The production of 316 grades are more economical however, hence its more widespread use.
Surgical grade steel is also called marine grade steel because its corrosion resistance also applies to saltwater. Hence its many uses that involve equipment and instruments that may be exposed to seas and oceans.