A direct merchant account is a virtual bank account used to accept and process credit card transactions. It is in the name of the merchant with funds deposited directly to the merchant's checking/savings account, that accepts ACH debits/credits.  The relationship is between the processor and merchant.

A 3rd party merchant account is an account in the name of a 3rd party, who is typically an aggregator (one who processes transactions for other merchants).  A 3rd party account may actually process for many different merchants.  The relationship is between the processor and the 3rd party account holder.  The 3rd party account holder processes for merchants and pays them each directly.  In many cases, there are substantial hold times and reserves involved, due to increased risk.  Fees are generally considerably higher than a direct merchant account.  The merchants using the account are generally NOT protected in this relationship.  3rd Party merchant accounts are typically reserved for those in a high risk category or those unable to obtain a direct merchant account.