Multiplayer in Mindustry works just as it does in many other sandbox games. To host a server, press 'host' in the menu while playing. Your server's IP can be found by googling my ip. To join a server, enter their IP in the 'add server' box and click the server. Both over-the-internet and LAN play is supported. For someone outside your network to connect to your games, port forwarding is required. The default port to use is 6567. Multiplayer is cross-platform, although the web version does not support hosting.
A dedicated server can be hosted on Windows, Linux, or Mac. Just download the 'mindustry-server-xx.jar' file from the downloads list and run 'java -jar mindustry-server-xx.jar'