Actually, Julian is correct here.
A CEO is the top management position within a corporation (and certain other organizational structures). A CEO must be hired by that organization (or you can declare yourself the CEO if it's your company).
Elections come into play only with a Board of Directors. In some situations, a CEO is entitled to a position on that Board. The members of the Board will elect a Chairman of the Board. While similarly powerful positions, and while a CEO may also serve as the Chairman of the Board, they are not one and the same.
To put it very simply, a CEO needs to be appointed or hired. Elections take place within a Board of Directors.
What can happen, however, is that a Board of Directors can vote to remove (basically to fire) a CEO if that individual is underperforming. However, a CEO will never run for an election against other competitors.
Hopefully that clears it up a bit.