The term close corporation is used in two related but different ways in corporate law. Sometimes when people use the term "close corporation" they are referring to the larger category of closely held corporations, i.e. those that are not publicly traded corporations. In a legal context, however, the term close corporation often refers to a narrower category of closely held corporations, those that have elected to be treated as statutory close corporations under corporate law. For example, Delaware General Corporation Law Subchapter XIV (Sections 341-356) contain a regime under which closely held corporations can elect to be treated as statutory "close corporations."
What's the difference between an ordinary closely held corporation and a close corporation under a statute like Delaware's Subchapter XIV? The main difference is that statutory close corporations can operate with a more informal or decentralized managment than the traditional board-centric corporation. Indeed, they can even be managed by the stockholders directly similar to the way a general partnership typically operates. Thus, the statutory close corporation has more flexibility than the standard corporation in how it will be managed.
In some states, such as Delaware, a corporation must explicitly opt into the special close corporation regime. In other words, a Delaware court likely would not apply the close corporation statute to a corporation that had not followed the procedure to become a close corporation. However, in other states courts will sometimes classify corporations as close corporations in the absence of an explicit election. Indeed, a New York court even did so to a Delaware corporation in Zion v. Kurtz.