What is a Shortlink URL? - Let's say your WordPress site was installed on a domain http://mysitename.com/. But let's say you also owned a domain name, http://short.net/ which pointed to http://mysitename.com/. Pretty Links's Shortlink URL feature, will let you use http://short.net/ as the URL used in your Pretty Links. So let's say you had a Pretty Links like http://mysitename.com/xyz, when you change the Shortlink URL setting in Pretty Links, the Pretty Links would then look like http://short.net/xyz.
Can I use any domain here? - Yes and No. As long as you own the other domain, and it points to your WordPress site where Pretty Links is installed, then it should work. You can't use a domain name that you do not own, and you cannot use a domain name you own that does not point to your WordPress site.
I own a second domain, how do I point it to my site? - Each web hosting company may handle this a bit differently so it is best to contact them for help with this. However, generally speaking you will want to setup your alternate URL as a Parked domain which uses the same directory/folder as your main WordPress site. The alternate domain will also need to use the same NS or A, AAAA records as your main domain. If you're unsure about how to do this, contact your web hosting company for assistance setting up the Parked domain and its DNS on your hosting account.
Alternatively if your domain provider supports wildcard domain forwarding, then this may be an even easier option. Wildcard just means that http://short.net/* will forward to http://mysitename.com/* (no matter what * is).