<i>The</i> 1st party is the platform vendor. So, Microsoft for PCs, Apple for Macs and iPhones, etc.<p><i>The</i> 2nd party is the customer, either you or the organisation you work for.<p><i>A</i> 3rd party is anyone else.<p>Choice of indefinite rather than definite article in the last case is deliberate.<p>The terms come from law: most contracts are between two entities ("the first party" and "the second party") and any other entities not party to the agreement are referred to as "third parties".