Because of the way people throw around the phrases "The Dow" and "Nasdaq," both terms have become synonymous with "the market," giving some people an inaccurate idea of what each term actually means. "The Dow" actually refers to the Dow Jones Industrial Average (DJIA), an important index that many people watch to get an indication of how well the overall stock market is performing. The Dow, or the DJIA, is not the same as Dow Jones and Company, the firm that publishes the Wall Street Journal. The gauge is one of many indexes owned by S&P Dow Jones Indices LLC, a joint venture of S&P Global (SPGI) and CME Group Inc.
Nasdaq is also a term that can refer to two different things: first, it is the National Association of Securities Dealers Automated Quotations exchange, the first electronic exchange that allowed investors to buy and sell stock on a computerized, speedy and transparent system, without the need for a physical trading floor. The second reference refers to an index. When you hear people say that the "the Nasdaq is up today," they are referring to the Nasdaq Composite Index, which, like the DJIA, is a statistical measure of a portion of the market.
Both the Dow and the Nasdaq, then, refer to an index, or an average of a bunch of numbers derived from the price movements of certain stocks. The DJIA tracks the performance of 30 different companies that are considered major players in their industries. The Nasdaq Composite, on the other hand, tracks approximately 4,000 stocks, all of which are traded on the Nasdaq exchange. The DJIA is composed mainly of companies found on the NYSE, with only a couple of Nasdaq-listed stocks such as Apple (AAPL), Intel (INTC), Cisco (CSCO), and Microsoft (MSFT).
Remember, although both "the Dow" and the "Nasdaq" refer to market indices, only the Nasdaq also refers to an exchange where investors can buy and sell stock. Furthermore, an investor can't trade the Dow or the Nasdaq indexes because they each represent merely a mathematical average that people use to try and make sense of the stock market. You can, however, purchase index funds which are a kind of mutual fund, or exchange traded funds which are securities that track these indexes.