Iji.
Edit: This is definitely one to play, not watch. Avoid any spoilers like the plague, or any information at all if possible.
+1. Iji is a fucking great game.
I guess theres always Xenomoviesaga, but your mileage may vary on the actual game/story of it (and it gets pretty weird, plus the 2nd ones gameplay kinda suffered a bit).
-1. Get Xenogears instead. Gameplay is better, the game has some world map funtimes to make it not seem/be quite as linear.
Also +1 for Final Fantasy Tactics, though that's only because it's my favorite game out of probably all the vidya I've played.
Will also +1 Analogue: A Hate Story. It's ppreeetty much a visual novel wherein you dig through the log entries of a dead ship. Will also recommend other stuff by the author: Digital: A Love Story, and sorry babe, it just ain't your story. They're fun times. They are, however, most definitely visual novels.
ARE you interested in visual novels? If so, look up Dischan - they've got
Dysfunctional Systems and Juniper's Knot, which are both short, and pretty good reads.
In the realm of more traditional vidya:
KOtoR - that is, the non-MMO Knights of the Old Republic games - have pretty good stories, both. KOtoR 1 I really liked; 2 is unfinished and thus kinda hacked up at the end, though there are supposedly mods for the PC version that restores some of the content.
+1 for Planescape: Torment. Supposedly one of the best stories in video games - I wouldn't know, personally, as I never got passed a very early bit in the beginning, before any of the good stuff happens. As mentioned, not the best for couch-stuff.
Child of Light is an interesting one, supposedly (it doesn't run well on my low-end PC). It's like a fairytale, dialogue is in rhyme, stuff like that.
The Yawhg is a sorta story-based game. It's a really simple one, though - don't expect there to be a lot of big twists or changes between games. Get some friends though, cuz' it's multiplayer! Or one-guy-controlling-four-guys-player. Or two-guys-controlling-two or whatever. Turnbased and stuff!
https://www.youtube.com/watch?v=4MGvyKG-I3U