Dans' wife pulled off an Initial Public Offering in front of the boss at the company party over the weekend. You just know that asshole is going to get a promotion now.
Incorporation is a legal status that allows (but by no means requires) a firm to issue stock. Moreover, once a corporation lists stock, it does not necessarily do so on a major exchange. Some corporations areclosely held, which means they have a small number of shareholders who are mostly affiliated with management; other corporations are "private," which means they have no stock issues at all, and control/shares of profits are determined contractually.
Some corporations have issues of stock, and that stock is traded, but it is not listed. Instead, it is traded on the "pink pages." Such companies are usually in a bad way, but not necessarily.
An IPO is the first issue of stock by a corporation THAT DOESN'T ALREADY have a listed stock. If a company is "taken private" (i.e., bought out by a PE fund and de-listed) then it can have another IPO (or "sponsored IPO"). Most likely, however, if a listed company will need to raise money on the stock market, it will have a "follow-on offering."