Question: Yesterday, I noticed that <https://hkps.pool.sks-keyservers.net/> showed a different page on my laptop than it did on my desktop PC.

This URL redirects to one of the many servers listed here:
<https://sks-keyservers.net/status/>

Unlike a normal URL redirection, the URL does not change, while the contents of the page do.

How exactly does this work?
How can it be set up?
Does it have anything to do with a [round-robin DNS system](https://en.wikipedia.org/wiki/Round_robin_dns)?

It also seems that it always shows the same "server" on a given PC once it has been accessed once from there. How does it do that?

<http://www.pool.ntp.org/en/> appears to be a similar system.

Answer: 
There are several ways to do this.  

Here is a very superficial way describe this. 

One common way is to have "<https://hkps.pool.sks-keyservers.net/>" point to a frontend with a webserver on it. 

The frontend then proxies your request to one of it's backend servers. There are many ways for the frontend server to choose a backend server `round robin` is one of them, `number of connections` is another.

When you return to "<https://hkps.pool.sks-keyservers.net/>" the frontend either remembers your IP and send you to the same backend or the backend gives you a cookie that the frontend reads that tells it you want to go to a particular backend.
This probably has nothing special to do with the DNS part of the connection.

Web servers can key off information your browser sends with it's request headers, e.g., "referer" tags which indicate you clicked on a link on another site to get here, or what's most likely in this case is the User-Agent string sent by your browser. Here's a couple examples of User-Agent strings:

Chrome version 32 running on some 64-bit flavor of Linux:

> 
> User-Agent: Mozilla/5.0 (X11; Linux x86\_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/32.0.1700.107 Safari/537.36
> 
> 
> 

or

Safari version 7.0.1 running on OS X 10.9.1:

> 
> User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10\_9\_1) AppleWebKit/537.73.11 (KHTML, like Gecko) Version/7.0.1 Safari/537.73.11
> 
> 
> 

This is also how web servers figure out if you are on a mobile device.

Here's Safari version 7.0 running on an iPad running IOS 7.0.4:

> 
> User-Agent: Mozilla/5.0 (iPad; CPU OS 7\_0\_4 like Mac OS X) AppleWebKit/537.51.1 (KHTML, like Gecko) Version/7.0 Mobile/11B554a Safari/9537.53
> 
> 
> 

The web server can use this information to display a different page if the designer made a conscious effort to do so. So you can get a different-looking page on Chrome versus Safari, as a simple example. But as you can see from the contents of the User-Agent strings, there's a lot of information there that a web designer can use to customize the resulting page you get back.