Star Trek
Star Trek is one of the classic nerd interests. If you are a nerd (And you are, otherwise why the fuck would you be on this site?) you are at least somewhat familiar with Star Trek. But then, you may have stumbled across this page by accident somehow, so here's the York Notes on Star Trek.
The Original Series
"Wagon Train to the stars". Pulpy adventure sci-fi, with fist-fights and gunfights and occasionally sword-fights. The Original Series isn't deep, folks. It's good, hammy fun, monster-of-the-week type stuff. The USS Enterprise is tasked by the Federation to seek out new worlds and boldly go where no man has gone before. James T. Kirk is the captain, and he fucks and fights his way to victory. Mr. Spock is cold and logical, Dr. McCoy is rash and emotional and Scotty gets shit done.
The Next Generation
Here's where it starts getting a little deeper, and a little darker. The USS Enterprise-D (usually just called Enterprise) is once again tasked with going where no-one has gone before, but this time around the problems are less likely to be solved in a single episode. Jean-Luc Picard is the captain, and he plots and negotiates his way to victory. Mr. Data is cold and unemotional (but he'd like to learn emotions), Riker and Worf have sex and punch things, respectively, and Geordi LaForge gets shit done.
Deep Space Nine
Unlike all the other series so far, Deep Space Nine primarily takes place on a space station - the titular Deep Space Nine, out near the borders of Federation Space. Said space station is near a recently-freed world and a wormhole, so all sorts of crazy shit goes down. It's a lot more political than other series (Though TNG and Voyager have their moments) and the last series to have Gene Roddenberry's involvement.
Voyager
Voyager is...well, it's controversial. Many people say it's the worst series, but it has its fans. The USS Voyager gets teleported over to the other side of the galaxy, and the plot of the series as a whole centres on its efforts to get back home. Like TNG and DS9, it's a character-driven drama just as often as it is a sci-fi adventure romp.
Enterprise
This is the closest series to being genuinely, objectively bad. It's a prequel to the rest, taking place on the first enterprise, before the Federation got a lot of shit figured out - so there's a lot of primitive versions of things from other series. At least the uniforms were pretty cool, in an air force sort of way.
So why should I care?
Because between them, these 5 TV series and their assorted spinoff movies, books, etc. provide inspiration for any sci-fi game you could care to run. If you want light-hearted action, look at the sort of things that happened in TOS or DS9 to get the crew into some dangerous situation. If you want a charismatic villain, look at Ghul Dukat or the Borg Queen. If you want moral issues and debates, look at the shit that happened to Voyager.