Despite our cultures emphasis on individuals, teamwork still pays a big part in our corporate environment. Only those who work for themselves, and by themselves, in cottage industries are truly on their own. All of the rest of us have to cope with other people to get things done. Customers rely on businesses to produce goods and services. In turn, bosses rely on their employees, and employees have to rely on their teammates to get projects done by the time they are due. If we all said, "The heck with teamwork," the entire economy would grind to a halt.
The key to teamwork is putting the needs of the project over one's own petty desires and cares. It is the goal that matters, not the individual. It is harder than it looks. Our culture seems to reward those who are not good team players, the mavericks and the shakers. Yet that is not necessarily true. If you are an up and coming executive, before you reach the top you have to learn to "play well with others." If you don't, you will be fired long before you hit a position of power. After you make it to the top, you may not look like a good team player, at least from the views of the trenches (the common worker), but that is just because your project goals are no longer the same as theirs.
There is a vast amount of difference between making a widget every hour, and how to make the most cost effective widget. The team making widgets on the assembly line are only concerned with putting the product together. The team supervising the assemblers are concerned with market placement, the competition, pricing and how to make the most profit. As one moves up though the ranks of a company, the team that you are on changes, and so does the goals that you are concerned with.
To be the most effective team member, learn what your team's goal is. Periodically, review it to see if it is still current (goals have a habit of changing over time). If you move to another team, learn what their goal is.
Also remember a large part of teamwork is getting along with other people. If a group of people is pulling in different directions, then less is going to be accomplished.
The importance of teamwork is that it allows us to accomplish more than we could by ourselves. It allows us to move mountains. If you doubt the importance of teamwork, just ask yourself where Bill Gates would be today without his team and the work they did.