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: 
Yes, all the pools of sks-keyservers.net are set up using a DNS round-robin. In terms of reverse proxies, it require all servers to have one enabled, and if you look at the rprox column at <https://sks-keyservers.net/status/> some blue flags are specifying servers with multiple servers in the backend in a clustered setup. 

The actual data for the round-robin is based on (i) hourly update run of the full pool (ii) authorative DNS server update the list of DNS records every 15 minutes. For the non-geographical pools a random selection is used in (ii), for the geographical pools (EU, NA, ... ) it is ranked by SRV record based on the description in <http://kfwebs.com/sks-keyservers-SRV.pdf>
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.