All through college I played Netrek, which was an awesome 2d 16-player (this was in the mid-nineties when Internet was young) reflexes/strategy game. You can learn more at netrek.org and I just found a Youtube video showing the netrek game as well.

Star Trek: Voyager - Elite Force

Star Fleet Battles of course!

One of the longest, most complicated games I've ever played. Expect a full day of playing with only a few players!

While not Trek-specific, I am a big fan of running Star Trek battles using Starmada.

There was an old ascii Trek game back in my TRS-80 days (that's pre-internet kids) that I used to play A LOT. It was single player, and a lot of fun.

Also played some StarFleet Battles about a year ago. Very fun.

Star Trek: Armada

Visual Star Trek

Super Star Trek