Basically put, the stock price of a company is a good measure of how the market as a whole thinks of the company's future, management expertise, and psrospects. A high stock price means that the market thinks it's a well managed company with a future and a good earnings flow, and a dropping stock price can mean the reverse.
This impacts the options available to management. Most especially, it acts as a gauge to how solid the company might be as a borrower. A company with no prospects is going to find it harder (and more expensive) to borrow money for expansion in the future, until the current situation (or perception thereof) turns around. This is one of the major reasons a firm's stock price gets so much management attention. Another is that hghly placed managers are likely to hold a bunch of stock themselves. This is another built-in motivator for management.
As to having to have a buyer if you're selling, you will. The company doesn't usually buy it back. Your broker does. All major brokerages have millions of shares in thousands of companies in their house accounts. When you buy or sell a share, the other half of the transaction is likely to be a broker.
Of course, at any given time, what you can buy a share for is a little more than what yuou could sell the same share for. The difference is called a "spread", and in addition to commissions, it's how brokerages make money. They get the ask, but they only pay the bid price.
Make sense?