Ownership of enough of a corporation that you have a voting majority. In privately-held corportations, controlling interest is a moot point -- the owner(s) controls the company. In publicly traded companies, voting power is proportional to the number of shares one owns; if a person owns more than 50% of a company, or owns a large enough fraction when a significant number of shares are not actively voted, that person has a controlling interest -- what action the board takes is entirely under their control.