A CEO is responsible for execution. Execution of a strategy, declaring a vision, setting the tone of the culture of all the workers, getting all hands working together and delivering value. Understand the market, their future in the market, the direction the company needs to move in, the strengths and weaknesses of themselves (and the company) and design strategies to mitigate weakness and risk.

In that role, maybe they make lots of decisions, maybe they write things like "Part Deux", or maybe they stand around an scream a lot. Whatever works; probably a little of all of the above, and a good CEO knows when to use which approach.

At some level they may also report to a board, who helps them, or replaces them, all based on their execution success. And at another level, they report to shareholders who do the same.