Stock is called "shares" because you have a share of the company - a piece of the pie.
If XYZ company has a million shares outstanding, and they're worth $60 apiece, the whole company is worth $60 million. If they split their stock 2 for 1, there will be *two* million shares of stock outstanding - but the company is still worth $60 million. That makes shares worth $30 apiece.
Companies split their stock primarily for psychological reasons. If someone is interested in investing in XYZ, but he only has $3000, he might buy 100 shares after the split. If he were to buys 50 shares at $60 apiece, he's not dealing in "round lots", and the brokerage company will charge him a lot more.
Many companies see their stock price go up after a split for that reason. Some people hear the news of a stock split, and illogically think "Oooh, they must be good", and are attracted to the company.
Some companies issue stock dividends. This is another kind of split. A 7% stock dividend means that for every 100 shares you owned before, you now own 107 shares. Some investors will sell those 7 shares, which means the price of the stock goes down. I think managements sometimes use this as a means of depressing the stock price so that they themselves can buy more stock on the cheap. Managements will also declare a stock dividend so that stockholders feel their stock is earning something; paying a cash dividend would deplete the company's cash reserves.
There are all sorts of splits and reverse splits. When AT&T was initially broken up, there were a number of "baby bell" companies formed, and stockholders got stock in each one of the baby bells. In a 3:2 reverse split, someone who has 300 shares before ends up with 200 shares after. In a 3:1 or a 7:1 split, the stockholder ends up with 3 times as many or 7 times as many shares.
If, due to a split or a stock dividend, you end up with fractional shares, you usually are given the opportunity to invest more to round up, or to sell your fractional share, without paying a broker's commission.