Hey me and my friend have been gaming for a while, and were looking for a good multiplayer game that preferably has good storyline and replayability.
List of games we have tried already:
Minecraft
Civ V
Cube World
TF2
Orcs Must DIe! 2
Magicka
Portal 2
Star Trek Online
Supreme Commander 2
Terraria
Sins of a Solar Empire: Trinity/Rebellion
Planetside 2
Want a good multiplayer game?
Found some good ones here, personally my favourite, here LMGTFY: http://www.lmgtfy.com/?q=Good+Multiplayer+games
Borderlands 2
Asking for good story AND multiplayer might be pushing it, a lot of games these days have one or the other really.
You could try DayZ or maybe Dark Souls or Dark Souls 2. The Dark Souls games aren't really for people that want to play multiplayer with friends though. Chivalry: Medieval Combat is good too.
I'd definitely say Borderlands 2. Especially if you could also get all of the DLC in the current sale. I bought it for a friend because I wanted him to try it out, and he loved it. We ended up playing around 1000+ hours together. Lots of replayability. Lots of loot. Good fun.
For games which you don't need to download, you can play Bloons TD 5 or Bloons TD Battles (you have to make a Ninjakiwi account first though) and Realm of the Mad God.
A downloadable game you can also try is League of Legends, or LoL
Borderlands 2 is the most incredible game for co-op play I have ever found. It is also just an all-around incredible game. Here is a list of why it is great for co-op off the top of my head:
1) Enough loot for everyone (especially if you strategically choose your character classes)
2) Classes that support each other
3) one difficulty level (no more arguing because that one dude can't take hard mode or even normal mode)
4) an incredible storyline (Handsome Jack is one of the best game villains I have ever known of)
5) funny (more than enough opportunities to create new inside jokes, and the game characters are funny also)