According to the FAQ (Frequently Asked Questions) list for the newsgroup rec.games.netrek:
"Netrek is a 16-player graphical real-time battle simulation with a Star Trek theme. The game is divided into two teams of 8 (or less), who dogfight each other and attempt to conquer each other's planets. There are several different types of ships, from fast, fragile scouts up to big, slow battleships; this allows a great deal of variance in play styles."
It is played over the Internet, against real human opponents. If you do not have a computer on the internet, or connected via SLIP or PPP,
you will not be able to play.
This game has a history and can actually be traced back through various ancestors to 1972. See the history compiled by Andy McFadden for more detailed information.
Up to 16 players, often widely separated geographically, connect to a central "server" running at some site. There are about 30 public servers in the world and of these maybe 10 are well known and popular.
The individual players uses a "client" program to connect to the server of their choice. Once in the game, the server receives commands from the client (and hence the player) and sends the positions and status of the other ships, planets, etc., to all players, several times per second. The net effect of all this is to create a virtual galaxy where everyone can see each other and interact, or to put it less academically, everyone plays in the same galaxy to try to take it over.