All stock exchanges and markets are called 'aftermarkets'. This is because when a stock is first issued, it's initial offering is called the 'primary' market. When you see 'aftermarket', this is referencing any of the exchanges and markets where regular trading takes place.

The major Exchanges:
New York Stock Exchange (NYSE)- the largest exchange listing over 4000 of the biggest publicly traded companies.
American Stock Exchange (AMEX)
(the vast majority of stocks are traded at the above 2)

3. Chicago Stock Exchange (mostly futures and options are traded here)
Each Exchange has it's own requirements and criteria for a stock to be listed. Some stocks are listed for trade on multiple exchanges, there are several other small regional exchanges, but the largest and best known companies are listed on the NYSE and AMEX, so most companies strive to be listed on these 2.

Over the Counter Market (OTC) - NASDAQ
Any company whose stocks are not listed for trade on an exchange are said to be Over the Counter or OTC. These are stocks which are bought and sold from the participating brokerages inventories. NASDAQ is the National Association of Securities Dealers Automated Quotation system. NASDAQ is the largest of the OTC markets. There is no single trading floor or location for these trades. These brokerages trade via telephone or computer using the NASDAQ computer system to buy and sell from their own inventories. The vast majority of stocks listed are on NASDAQ, there are over 17,000 stocks traded in the OTC market.
NASDAQ considers itself on par with the larger more traditional Stock Exchanges, because it includes some of the largest public companies in the market today. Intel, Microsoft, Apple Computer, MCI are among the major corporations that are traded on NASDAQ, in addition to most of the Internet stocks that are on the market. The NASDAQ system is faster because it's more reliant on technology than the more traditional Exchanges systems of brokers on the trading floor.
There are less regulations in the OTC market than in the Exchanges, in that NASD does not set a minimum or maximum limit on commissions, markups or markdowns.

Initial Public Offerings- IPO's
Just a quick word on these. IPO's are the stock issued by a company going public for the first time. These stocks are first sold by an 'underwriter', a firm which sets an initial price and markets the stock to the public. You can only purchase the stock at IPO prices if you have an existing account with the underwriting brokerage. Most people don't have a chance to participate in this 'primary' market IPO sale. They must wait til the stocks are publicly traded on an Exchange or OTC before they can buy. Thus the price you see on an IPO is not usually a price you can get in on. Though if you could, for a hot new company, it's probably a money maker:). Not all IPO's go up in value however, so do your research first before jumping in on one of these.