Surgical Stainless Steel
The term surgical stainless steel is primarily a marketing term, and does not refer to one type (or grade) of steel. In fact, your cutlery at home and your fancy stainless steel watch are both likely to be of the same quality as "surgical stainless steel."
The 'surgical' probably refers to the belief that these types of steel are well-suited for making surgical implants and equipment; they are easy to clean and sterilize, strong, and corrosion-resistant. However, some patients may have negative immunoresponses with nickel, commonly mixed with steel to make an alloy.
Although there are many variations in the recipes, there are two main types of stainless steel - martensitic and austenitic.
Most surgical and piercing equipment, like needles and clamps are made from martensitic steel - it is much harder than austenitic steel and easier to keep sharp. Depending on the type of equipment, the alloy recipe is varied slightly to get more sharpness, or strength.
Implants and equipment that are put under pressure (bone fixation screws, prostheses, body piercing jewelry) are made out of austenitic steel, often 316L and 316LVM, because it is less brittle and far more corrosion-resistant.
Stainless steel (types 316L and 316LVM) is still the most commonly used jewelry material in the piercing industry, although titanium and "nickel-free steels" are gaining popularity.