From the onset, let's be clear about the definitions.

A desktop PC or personal computer is usually powered by an x86 (either Intel or AMD) processor, runs Microsoft's Windows operating system, is connected to a monitor and the mains and is usually fixed. Yes, there are a myriad of other combinations (ARM-based and Android or the endless flavours of Linux), but we'll stick to those for now.
A workstation PC is a subset of the desktop PC family. They are primarily used by engineers (in the widest interpretation of the term), analysts, designers, content creation professionals, developers and anyone that requires data manipulation with a strong visual slant.
A Gaming PC is the consumer alter-ego of the workstation PC and, as its name implies, is used mostly for gaming. However, their sheer performance and value-for-money ratio have made – for some – a compelling alternative to their workstation siblings.