A third party check is one signed over by the intended recipient to someone else to deposit or cash. The original recipient must endorse the check first, ideally signing it over as "pay to the order of" followed by the name of the person he's giving to. That recipient then endorses it and deposits it into his own account. These checks are usually harder to deposit or cash than regular two-party checks because they represent a higher risk to the bank.
Banks might not let you deposit third party checks at ATMs. At TD Bank, for example, you'll have to see a teller and fill out an Endorsement Authorization & Release Form. You may also have an extended hold on the funds until the check clears the issuing bank, depending on your bank's policy. Moreover, third party checks can be difficult for the recipient to collect upon should the check be returned for insufficient funds. The claim would be not against the person who gave you the check, but the one who wrote it.
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Third party checks are risky to accept for a number of reasons. In a guide on money laundering designed to provide advice to money services businesses, the U.S. Treasury Department's Financial Crimes Enforcement Network said that cashing a large number of third party checks, or paying for services with third party checks, might be a sign of a money laundering scheme. If you're given such a check by someone you don't know and trust, there's a chance that you'll be an unwitting party in check fraud.