Stainless steel is a steel alloy which is highly resistant to staining, rust, oxidization, corrosion, and tarnishing. It gains this quality from the chromium in it, which must be at least 10.5% of the alloy by weight. Because it is a type of steel, it is also very strong.
Steel is simply a combination of iron with some other element to make it harder and stronger, usually carbon.
For jewelry-making applications, surgical steel is effectively the same as stainless steel.