Corporations not allowed to issue stock are referred to as "non-stock" corporations; those who are considered the owners of a non-stock corporation are persons (or other entities) who have obtained membership in the corporation and are referred to as a "member" of the corporation.Corporations chartered in regions where they are distinguished by whether they are allowed to be for profit or not are referred to as "for profit" and "not-for-profit" corporations, respectively.
There is some overlap between stock/non-stock and for-profit/not-for-profit in that not-for-profit corporations are always non-stock as well.
A for-profit corporation is almost always a stock corporation, but some for-profit corporations may choose to be non-stock.
In American English, the word corporation is most often used to describe large business corporations.
In British English and in the Commonwealth countries, the term company is more widely used to describe the same sort of entity while the word corporation encompasses all incorporated entities.
In American English, the word company can include entities such as partnerships that would not be referred to as companies in British English as they are not a separate legal entity.