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Connecting a GPU by Ethernet is like connecting your PS3 controller to the PS/2 keyboard port: sounds like it's trying to fix a problem but the solution turns out to be completely outlandish and impractical. (Granted, if you can pull it off, you'll gain more popularity as that one crazy guy.)

That said, I advise you not to try to do this because the interface simply doesn't exist. You'd have to go through a lot of hoops to get your card to accept low-level commands through high-level interfaces and then bring back a video output. The latency would be absolutely horrible, unless you are willing to pay thousands and thousands of dollars for an enterprise-grade solution that lets you do this for one reason or another.

However, don't fret. You may still be able to connect your GPU to your laptop. For instance, if you have an ExpressCard slot, you can use an ExpressCard to PCIe adapter (along with a power supply unit) to have a seamless experience.

However, there may be compatibility issues due to the operating system's inability to set up resources for the PCI connection to the GPU, so browse around and make sure it's compatible before you buy anything. Consult guides such as PCWorld's eGPU tutorial to see if you have everything you need to use a desktop GPU with your laptop.

Regardless, do not expect 100% performance. Remember, it's a laptop.

Probably not at commodity pricing for another few years. The current offerings are all high-end, using 10/40 Gbps connections and Nvidia Tesla cards. Even then, it's not directly usable for gaming or graphics, but more for CUDA processing.

e.g. http://www.chelsio.com/gpudirect-rdma/

Not with ethernet, but with PCIe and Thunderbolt. This article breaks down the external GPU (eGPU) landscape well.

A number of companies sell PCIe/Thunderbolt enclosures. Some are limited by Thunderbolt's power, some have their own power.

• 
  MSI GUS II using Thunderbolt and limited to 150W.


• 
  Akitio Thunder2 PCIe Box using Thunderbolt 2, but only provides 25W.


• 
  Netstor sells a number of externally powered PCIe expansion systems which run on Thunderbolt.


• The Razor Core is a Thunderbolt 3 external enclosure handling up to 375W.

There's even a DIY kit.

Not by Ethernet, but PC Mag ran a story last Sept on using a gaming card with a laptop connected via PCIe adapter card, Express Card, or Thunderbolt for Macbooks.

http://www.pcworld.com/article/2984716/laptop-computers/how-to-transform-your-laptop-into-a-gaming-powerhouse-with-an-external-graphics-card.html

The author uses a 2011 Thinkpad and under $200 GPU and shows benchmarks for some modern games. I'd summarize him as pleased he could run them at all for a small expenditure (not a full new gaming box). Benchmarks are provided to make your own judgements.

The closest you can come to what you want is, if you're gaming via Steam, to use the in home streaming option to use your network to route the display from the a gaming desktop to your laptop and user input from the laptop to the desktop running the game. This would require a full desktop somewhere in your home instead of just a laptop + external GPU; but you can hide it out of the way somewhere and only interact with your laptop.

If you have no technical skills, then the easiest way to break this down is simply: does your graphics card have an Ethernet port? The next question being "Can I get an adapter to make it so?" As far as I know, there is no adapter capable of adding an Ethernet port to a graphics card.

If you simply want to beef up your laptop with an external graphics card, there does happen to be a solution for that, though it's obviously not cheap. A better solution would be buy or build a gaming specific machine, based on the specs required for the specific games and types of games that you want to play, for the number of years that you think the machine will last/stay relevant. Maximum PC usually has good articles recommending gaming rigs, I suggest you check out their build pages. They usually have 3 different tiers with parts lists and links. mid range linked earlier.

If you insist on using the adapter, here is the tech: This device can use a Expresscard slot or a mini PCIe slot. The Expresscard can be found on newer laptops, and is externally accessible. The mini PCIe slot means you'd have to do surgery to your laptop and route a cable, assuming that your laptop had such a card slot.

I believe VirtualGL might help you with this - it allows you to run the graphically-intensive program on device and view the results on another.

I doubt you'd be able to run just a graphics card by itself via ethernet. But you could connect to another PC using some kind of remote desktop application (VNC, etc).

There has been rumblings about being able to connect an external GPU via Thunderbolt or USB 3.1, and AMD says they want to make a standard for it:
http://arstechnica.com/gaming/2016/03/amd-wants-to-standardise-the-external-gpu/

It's 2018 and eGPUs are a widely available thing now, it's still missing in these answers so I want to leave it as an answer in case someone isn't aware of those great things:

(Amazon Link to the above product)

These devices fit a GPU and a power supply (most of them have them built-in, the above one has a 400W one built-in to power the GPU). It's using Thunderbolt 3 which has a bandwidth of up to 40Gbit/s, offers 18V @ 550mA, can drive multiple 4K displays, and much more fancy things.

This would be your best solution. It's simple, there's cheaper ones too, and it guarantees you a perfect system integration and performance (you can even run a GTX 1080 inside these things).

As already pointed out, a solution involving ethernet has several flaws, the biggest being that your laptop's port most likely doesn't support more than 1Gbit/s.