What is an eGPU?
GPU stands for Graphics Processing Unit, which is more commonly referred to as a Video Card or Graphics Chip. The “e” prefix stands for “external”. In short, an eGPU is the act of hooking up a desktop video card to a laptop, or a SFF system lacking actual desktop-sized slots (such as an Intel NUC).

But why?! Wouldn’t a desktop make more sense? Wouldn’t a desktop erform better?
Why not? Maybe. Most of the time.

To elaborate:
Not all people want a desktop. Desktops are (typically) large, (typically) bulky and (by definition) immobile. There is convenience in having your own system with you on the road, while still being able to game in the comfort of your own home, without having to sync any data, or switch systems. One system is convenient, two are less so.

That said, if there is no particular wish to use a single machine, or if a laptop is not needed/desired in the first place, a desktop machine is undeniably superior (and often cheaper, if we compare the price of a laptop+eGPU setup to the price of a desktop built from scratch). There are a few considerations that make eGPUs desirable: Already owning a laptop (often a high-end one, for example, due to the requirements of an occupation) and wishing to be able to game on it, having an older laptop that could use a little boost in the arm in the graphics department but is otherwise perfectly usable, and saving space (because an eGPU plus a laptop take up very little space and are easier to fit into a small apartment or a dorm).

A desktop system with a near top-of-the-line desktop CPU (like the Intel i7 6700K or i5 6600K) and a given video card will nearly always outperform a laptop (no matter how high end the laptop is) with the same video card connected as an eGPU. This is an undeniable fact. However, eGPU performance can range from ~70% to ~95% of the equivalent desktop performance (depending on how the eGPU is connected to the laptop, whether you are using the internal or an external display, the game in question, the resolution you are at, as well as the frame rate you are getting), so the performance is still definitely there and is definitely viable………………………………………….   More

Source – https://www.reddit.com/r/eGPU/comments/5jpf2x/diy_egpu_101_introduction_to_egpu/