Stocks are traded on exchanges, like the NYSE, NASDAQ and AMEX. There are two main types of exchanges, physical and virtual.

The physical exchanges are the ones pictured in movies and on CNBC, where the crazy guys with the blue jackets wave about pieces of paper and yell out prices.
The virtual exchanges are actually linked computer networks, and the entire trading process takes place electronically.

The reason stocks are traded on exchanges is that it is the best way to facilitate transactions. It would be fairly slow to place classified ads in your local paper to buy or sell stock. The exchanges act as intermediaries between the buyers and sellers. Typically, electronic exchanges are more efficient, which is why even the face-to-face exchanges normally have electronic transaction services.