When referring to a company, a third-party is another company that has no affiliation or association with the current company. For example, IBM is a third-party company to Computer Hope. Computer Hope offers support for all third-party computer companies but is not affiliated with any other computer company.
With software, a third-party program is any software, plug-in, or mod that is not included with a software program. For example, Norton Anti-Virus is a third-party program to Microsoft Windows since it is not included with Windows.
Finally, with hardware, a third-party hardware upgrade is any upgrade or addition that is added to an OEM computer. For example, if you upgraded a Dell computer with new memory, the memory would be considered third-party hardware.