The most important tasks of a CEO
At any company the CEO is the most important person, they are after all the one who is running the show. While the CEO is ultimately responsible for everything that the company does there are some tasks that will occupy the majority of his time. While the specifics will vary from company to company there are some things that are common to all CEO's.
The most important task of the CEO by far is devising strategy. This is basically doing the long term planning for the company. Most people working at a business tend to get caught up in the day to day operations and these are important. However there has to be somebody thinking about the direction the company should go in the future and how to go about doing it. This is a job for the CEO and really the most important, almost everything else can be delegated but this can't.
Another important task for the CEO is to build the company culture. All businesses have their own culture, if you ever change jobs you will notice this very quickly. This culture is an important part of the way that the business operates and it does not happen by accident. Rather the corporate culture is largely set by the CEO who makes decisions on what kind of business he wants to have. This culture will affect everything that the company does so it is something that requires some thought. This is another of the things that the CEO really can't delegate to somebody else; in large part the company is going to follow the lead that he sets.
The CEO is also responsible for team building. He is probably not responsible for the hiring of most employees, likely only the other senior executives. However the way that CEO interacts with the employees will go a long way to determining how well they work as a team. If the CEO makes building a strong team an important part of his job he will almost certainly get one. However if the CEO is not actively working at building a team he will find that he gets a group of employees who don't work well together.
All companies have to make decisions about how they are going to spend money and ultimately this is a responsibility of the CEO. He probably won't be involved in the day to day expenses but he will be involved the decisions about capital allocation. He will have to make decisions about whether new money is going to be invested in a new plant or in a better distribution system for example. These are important decisions that will have a huge impact on the company's future. That means that capital allocation is closely related to strategic planning which is another of the CEO's responsibilities.