Do you remember supply and demand? I will illustrate by comparing a company's stock to a city. If there are more people moving away from a city do to a STORM than there are moving to the city, what would happen to the value of the houses in this particular city? Since more sellers are selling than there are buyers buying, the sellers will have to lower the price to compete with each other. This would be a buyers market. The buyer has the upper hand.
Now, picture the STORM I mentioned, instead of the storm being a hurricane and the city being a city, the storm is now a newspaper article entitled, "Intel Misses Wall Streets Expectations
buy a penny," and the city is now a society of people who own stock in Intel.
Let's say you and I own a pizza shop we invented, Atari's Pizza. We have 5 tables and two grills for a total of 800 square ft. However we get very busy @ night, we'll do $20,000 in sales a night. We ask ourselves one night after work, "imagine if every city in the country had an Atari's pizza"
In order to make that happen, you will need $$$$$. In order to get the $$$$, you will offer the world a small stake in Atari's and decide to sell 2million shares for 20$ a share. This is called an IPO (initial public offering), so to answer part of your question, you (Atari's) set the initial price, hence, "initial offering."
Ok now the offering is over and you have 2million times 20$. Let's say that much money is enough to get you twenty stores up and running right away. Now how many stores will you have in 20 years? If each store gives you 20k a night, this is a lot of money. It will be enough to add 5 new Atari's every year in different towns.
Think about it now, you sold 2 million shares to the public for twenty bucks and that was enough to get 20 new stores up and running right away. Also it was enough to sustain 5 new stores a year.
Now for the arithmetic, 20 stores (Stock Price 20$)+ 5 new stores a year for twenty years = 120 stores
Hypothetically, the price of the stock would be @ $120 because if investors were willing to pay 20 dollars when Atari's only had 20 stores then they would pay 40 dollars when Atari's had 40 stores as long as Atari's rate of growth didn't change, all else equal in other words.
This also sets the price or stockholders equity. What if Pizza Hut wanted to buy Atari's. They would be buying from the stock holders not the original two owners. That supports why 120$ a share is sufficient for 120 stores. That is what Ataris' is worth.