No router is needed if you are only trying to use it from the same computer that Superpowers is running on you can double click "My Server" in the app, or open your browser to http://127.0.0.1:4237 from that computer
Otherwise, to access it from another computer, you will still keep "My Server" in the app on the server as the 127.0.0.1:4237 IP, but you will need some kind of modem or router or switch attached to that computer to another:
1. Switch (local network) - There's no DHCP in the average switch-only device so you assign the IP addresses to any device connected manually, then open the browser on other computers to http://ip.of.the.server:4237/ (or add that as a server on the app from those computers)
This only works in a local network though, not through the Internet. Still, could be cool having a 24 computer LAN party!
2. Modem direct connection (online) - Connect the computer running Superpowers directly to the modem. Then connect from other Internet-connected devices again by going to http://ip.of.the.server:4237/ with a browser or adding it as a server on the other computer.
If your provider gives more than one public IP you can put a switch between the modem and router and then give it one of the public IPs, not sure if it would be assigned automatically from the modem or something you manually assign (this probably would not work this way if, like many, your modem and router are combined such as an ADSL router, not sure what you would do there sorry!)
Putting your desktop directly on the Internet is dangerous! Routers are handy for hiding your actual computer and their firewalls for blocking hazardous traffic. Putting a personal computer directly on the web like this is not a great option generally.
3. Router (online/local) - Connect your desktop running Superpowers to the router as normal, then port-forward 4237 and 4238 on the router settings to the IP of your desktop in the local network eg: 192.168.0.x (I think for just TCP, but maybe also UDP?).
Then you can find a "what is my ip" website on that computer and write down its public/Internet IP, then from the other computers you can go to http://your.public.ip.address:4237/ to connect
Local users (devices on the same router connection) should generally be able to connect directly to the http://192.168.0.x:4237/ IP though too.
4. Direct connection to another computer: You could get a cross-over cable (might not be needed if both computers have gigabit ports) to link two computers directly together with one ethernet cable. The server computer still goes to 127.0.0.1 while the other computer goes to the IP of the server (remember the :4237)
Also, I believe you will need to set the "Server password" in the app's SETTINGS tab on the desktop running the Superpowers server in order to use it remotely!
Hope that all helps! :)