Listing requirements are criteria a stock must meet in order to be traded on an exchange such as NYSE, Nasdaq, or AMEX. Each exchange sets its own requirements and failure to meet them can result in removal.
Stock exchanges want to make sure that the companies trading on their platforms meet certain standards so that investors will have confidence and continue to trade using the exchange. A first round of weeding-out will already have been done for stock pickers. Meeting the listing requirements is a stamp of approval for the companies trading on the exchanges.
The NYSE, for instance, monitors companies for corporate compliance and financial compliance. It reviews companies' practices for accountability and transparency, as well as items such as cash flow, market capitalization, and trading volume. It keeps an ongoing watch and can delist stocks not meeting the given criteria.
Listing requirements can be found on the exchanges' websites.
Investors looking at stocks trading over the counter OTC (not on an exchange) don't have the benefit of the initial vetting provided by the exchanges.
Listing requirements can be changed, of course, with approval (in the U.S.) from the Securities and Exchange Commission, which reviews proposed changes and collects public comments.