A stock is a share in a company, if there are a million shares then 1 share is a millionth of the company. Sometimes companies pay dividends (money) to shareholders every so often, if they don't people expect that at some point in the future they will so they are interested in buying. The price of a share is determined by how much people are willing to pay for it, a higher share price means a business has an easier time getting loans or it can issue (create) new shares to sell. Issuing shares dilutes the stock, if you have 1 million shares and issue 1 million more the value of 1 share halves, since the owners are also shareholders they try to limit this or they create a plan to only sell the new shares at a certain price.
Take a look at these key statistics for Ford Motor Company, at the top you can see their "market cap", this is the value of the entire company, the number of shares multiplied by the share price, the price is currently X, on the bottom right you can see the number of "shares outstanding" (roughly the number of shares), try the math yourself.
http://finance.yahoo.com/q/ks?s=F+Key+Statistics
On the bottom right you can also see information about dividends, the dividend yield is 3.8% which means in a year they will receive 3.8% of the value of the share at its current price. Or 15 cents per share each quarter (quarter of a year). See if you can do the math yourself.
http://www.nasdaq.com/symbol/f/dividend-history
Why does the company pay dividends? Because shareholders are owners of the company, as owners they often get a vote in how the company is run, though some companies sell different kinds of shares whose holders do not get a vote or they have other decision making processes.
In a normal marketplace like the one where you buy candy, people offer to sell things for a certain price and buyers have an idea in their mind of what they are willing to pay for it. In other markets buyers make bids and sellers make offers, each adjust their prices depending on the bids and offers others are making.
In the stock market there might be 1000s of different buyers and sellers so you need a middleman (or a middlewoman) to streamline the process. Market makers bid or ask to buy shares for a certain price from these large numbers of buyers and sellers. If no one is willing to buy shares they lower the price, if no one is willing to sell shares they raise the price. The difference between the price they buy and sell is called the bid-ask spread, multiple market makers compete with each other and drive the bid-ask spread down, in the past before computerization the bid-ask spread was quite high but now it is very low.
The way that all this is done is complicated because of all the red tape. Though the reason for all this red tape is simple, security measures, one does not simply trust another person with their money, in fact you can't even trust yourself with your own money as it is easy to make mistakes. So there are a lot of regulations to prevent fraud and prevent people making (huge) mistakes.
When a company wants to go public it goes to an investment bank, they negotiate a deal to divide the ownership of the company into shares and sell a number of them to the investment bank who will in turn sell them on a stock exchange, like the New York stock exchange. This is called an initial public offering (IPO).
The way they do this is complicated and full of red tape but generally the company sells shares to the investment bank who in turn sells them to the market.
http://en.wikipedia.org/wiki/Initial_public_offering#Quiet_period