In addition to the above answer, several of the video games are captured in other mediums:
Dark Forces/Jedi Knight
This series has it's own set of novellas covering the story: Imperial Soldier, Rebel Agent, and Jedi Knight. Following this, this character made more appearances in the larger EU (primarily in the New Jedi Order series of books). These cover enough of the events of the associated games that you won't miss anything. There is also an audio drama of these events, though I have not heard it so I can't speak to it's quality.
Tales of The Jedi Comics/Audio Drama
The Knights of the Old Republic I & II, and The Old Republic video games are rarely referenced elsewhere in the expanded universe. Primarily the references to this time period (such as Exar Kun) come from the Tales of the Jedi and Knights of the Old Republic comic series. These events prequel both games and are covered very well in comic form but also in a fully voice acted Audio Drama.
Note that this is the original trade paperback called "Knights of the Old Republic" and not the Knights of the Old Republic Dark Horse series released sometime later. That series also has very little to do with the story of the games, aside from a similar setting, similar locations, and occasionally cross-over characters.
X-Wing & TIE Fighter
Most of the events of these games are irrelevant to the larger universe, with a few exceptions (like the Redemption Scenario, and later Farlanders appearance in the New Jedi Order book series). However, both games have expanded story in the form of their original manuals. X-Wing has "The Farlander Papers" which is a small novella detailing his origin and back story, and TIE Fighter had "The Stele Chronicles" which detailed Maarek Stele and his pursuit to become a TIE Fighter Pilot.
Both of these stories were later expanded in their Official Strategy Guides, which detail every mission in a first person narrative and also post mission conversations with other characters and some pre-mission setting building too that does a wonderful job of putting the missions in context.
X-Wing vs. TIE Fighter had no real canon story at the time of it's launch, it was mostly just a multiplayer game. The expansion, Balance of Power, added in a story-based campaign, but there was, as far as I know, nothing detailing the story, and it was woefully devoid of characters or plot, unlike the X-Wing and especially the TIE Fighter games.
X-Wing Alliance had some additional story and character details in the manual, as well there was an in-game Email system that detailed the results of certain missions and other on-goings in the galaxy relative to the story that you may be able to read up on. But much of this game was cross-over with other events - for example the Executor raiding the smuggler asteroid detail was covered in Shadows of the Empire, and the Battle of Endor, of course, was covered in Return of the Jedi.
Star Wars Galaxies
Star Wars Galaxies had it's own story, The Ruins of Dantooine, published that details some of the events in the game. I got the impression this was supposed to be a longer series, but this is the only book I know of published around these particular events. Some books, like Death Troopers, had some carry over into the Star Wars Galaxies game, but I don't think that's what you're looking for.
Shadows of the Empire
This series has both a comic book series, and a novelization of the events that take place inside it. Unlike most of the other Star Wars games, this one has a firmer place in the Star Wars Expanded Universe, in that the events here are referenced frequently by other sources.
Things to Avoid
As I mentioned before, the Knights of the Old Republic comic series by Dark Horse has nothing to do with the game. The Rogue Squadron comics and books have very little to do with the games (though there are some occasional references and cross-overs). The controversial Republic Commando books have absolutely nothing to do with the game, though Imperial Commando deals with some of the events that take place after the game.
Things Not To Worry About
Obviously the Star Wars Lego games are spoofs, and should not be taken seriously. But the same should be said for the Rebel Assault and Rebel Assault II games. These games often take extreme liberty with the Star Wars universe (Rebel Assault, for example, has a rebel base on Tatooine) that cannot be rectified with the existing Expanded Universe canon.
The same can be true for most of the games that cover the same time period as the movies, for example the Battlefront Games often have different versions of key battles, like the Battle of Endor, where anyone can blow up the bunker.
Star Wars Rebellion can be written off for having absolutely no story worth speaking of. Some of the other strategy games have quite a bit of story (some of it very interesting) that is told only through the game, sadly.
