What the role means at a US firm
A senior associate is an attorney with enough years of practice to run matters with limited supervision and to supervise junior attorneys. In class year terms, that is generally a mid-career or later associate, though firms draw the line in different places. Some firms also use titles such as counsel or income partner for the step beyond senior associate.
At firms of 5 to 200 attorneys, the role is rarely just a bigger version of the associate role. It sits close to the partnership decision, and both sides are usually thinking about it.