Necessary Ports
In order to play Cosmic Supremacy your computer needs to have permission to open ports 4710-4799 (depending on the galaxy you play in) and port 6789 to the outside. This will allow the game to run in Direct-Connection mode, which is the preferred connection method.

Using Http-Tunneling
If that is not possible because you are sitting behind a firewall or a router, Cosmic Supremacy offers you to route its packets through http-port 80, using a technique named Http-Tunneling. This port is usually used to surf the web. You can activate this mode by clicking on 'Change Connection Settings' in the Progress-Dialog at game-startup, and ticking 'I Am Behind a Firewall'.

Playing via a Web-Proxy-Server
In rare occasions it is possible that even http-port 80 is closed. This is a likely scenario when you are working at a company that closed all incoming and outgoing connections. Usually the only door to the outside for employees is a so-called web-proxy. That means if you try to surf the web, you don't send requests to the web-site directly, but you send them to a web-proxy server instead, which will check if the request is valid, and then forward it to the desired web-site.

Cosmic Supremacy manages to by-pass this hurdle too, by routing its traffic via that web-proxy. However you need to find out the address and port of your local web-proxy server. Cosmic Supremacy will try to automatically detect those settings, but if it fails you need to enter them manually.

An example of a web-proxy server address could be proxy.provider.com at port 8080

If you are located in a company or school and need to use a proxy server, it is very likely that they will need you to use a login and password to authorize yourself to use that proxy-server. The connection settings dialog offers input-fields for that, too. The best place to find out about all that information is the web-browser that you use every day. Check for 'proxy settings' and basically copy these settings into the Cosmic Supremacy connection settings.