@Byshop: yeah sandbox games like minecraft don't need a story at all but if a game is made with a narrative and it's written well, I agree it can made a mediocre game amazing, and even make us overlook some issues would otherwise be obvious

For me it depends on the kind of game but story definitely matters, character development and having a reason to keep playing is most important to really get engaged.
Also having a great story really adds to replay value.

I guess I'm level 0!
Haven't played any, I was more into the Final Fantasy's and those types.

Where should I start to begin playing these games? I've definitely thought about getting into it recently.

Got to be Runescape for sure, spent years on that game back in the 2004 days, dial up also!

I did move onto Maplestory though, that game was amazing!

Other mentions would be Fly For Fun (Flyff) and Puzzle Pirates :)