Surgical stainless steel is a variation of steel consisting of an alloy of chromium (12-20%), molybdenum (0.2-3%), and sometimes nickel (8-12%).
The chromium gives the metal its sheen, scratch-resistance and corrosion resistance. The molybdenum gives corrosion-resistance, and helps maintaining a cutting edge.
Although there are myriads of variations in the recipes, there are two main varieties of stainless steel; martensitic and austenitic, see the stainless steel article.
The 'surgical' refers to the fact that these types of steel were once thought to be suited for making surgical implants and equipment: they are easy to clean and sterilize, strong and corrosion-resistant. However, due to immune response issues with nickel, steel has been mostly replaced by titanium in procedures that require a metal implant. Many piercists claim to use "implant grade" steel still, but the term is erroneous.
Most surgical equipment is made out of 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 is 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.