Due diligence is research and analysis of a company or organization in preparation for a business transaction, such as a purchase of a business or securities. In the context of a business purchase, buyers conduct due diligence to i) verify the accuracy of financial statements and representations; ii) discover problems; and iii) estimate what a business is worth. Here is an example of a due diligence checklist that could be used by a buyer: http://www.accountingtools.com/due-diligence-checklist.
Sellers also conduct due diligence to determine whether buyers i) are honest and competent; and ii) have the funds needed to buy and operate the business. This is especially important if a seller agrees to accept notes or deferred payments from a buyer.