Due diligence can have several meanings, depending on context. Generally it is the process of a prosepective buyer of shares or acquirer of a company confirming that the company has what it claims to have and there are not any hidden facts or potential liabilities.
If a company is selling shares, it may want to make sure that the buyers have done, or had an opportunity to conduct due diligence so they can't later say they were mislead. And the company may want to make sure the buyer is who he or she or it says it is and meets the tests of the offering.