Oh, and also, KEYDROP!
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2D Indie Platformers

• Braid
• Blocks That Matter
• Mark of the Ninja
• Terraria

Well, you didn't mention the budget, so these suggestions are from every price range.

Story:
Life Strange

Action:
Dragon Ball Xenoverse
DMC Devil May Cry
Castlevania Lords Of Shadow 1
Castlevania Lords Of Shadow 2
Dark Souls 1
Dark Souls 2: Scholar of the First Sin

Point-n-Click:
Deponia: The Complete Journey

Story Shooter? Try Metro 2033 (Redux) !

Alan Wake, Bastion, Transistor, Borderlands 2, Brothers - a tale of two sons, Botanicula, The Cave, Contrast, Darksiders series, Enslaved - Odyssey to West, Fable, Shadow of Mordor, Murdered Soul Suspect, Prototype Series, Sleeping Dogs, South Park SoT, Tiny and Big, Tomb Raider, Trine Series, The Wolf Among Us, Spec ops the line, Max Payne 3, FEAR 1, Binary Domain. Take a look. Damn, my fingers...

2D Platformers:
The Swapper is AMAZING, more of a story-driven adventure-puzzle.
Nihilumbra ♥
Max: The Curse of Brotherhood It really is 2D, but the devs do a fantastic job of making it seem like so much more!
Stick it to the Man!
NyxQuest
Out There Somewhere
All of the Trine, Oddworld, and Giana Sisters games

I have reviews of the first 4 if you're interested.

Story-driven Action-Adventures:
Tomb Raider
ENSLAVED: Odyssey to the West (have review)

and of course there is The Walking Dead Season 1 and Season 2

Bat Rats whops.. i mean Bad Rats

I hate video games

Oops, I forgot Freedom Fall. I had a lot of fun with this, I just wish that it was longer!

Story-Driven Shooters, like Half-Life or BioShock

• Spec Ops: The Line, Alice Madness Returns (not a shooter but platformer), Metal Gear Rising: Revengeance

Indies, any kind of indies can enter here, but I specially love 2D Indie Platformers

• Bad Bots, Bleed, Sang-Froid Tales of Werewolves, Audiosurf 2, Distance, The Bridge, Giana Sisters, Skullgirls, Jets'n'Guns Gold, Receiver

Point-n-Click, like Little Inferno or Grim Fandango

• Broken Sword, Dreamfall Chapters, Gray Matter,

Action, like the Batman Arkham series

• Remember Me, Serious Sam

Children friendly, like POSTAL 2, Manhunt or Hatred

• Master Reboot (it is kinda creepy), The Darkness II, Rise of the Triad

But any genere of videogames can enter here!

• PAC-MAN Championship Edition DX+, The Stanley Parable

The last Good game i played was Skyborn .

its an indie RPG maker game ...
Kind of short ... generic RPG maker game , but i really liked the story and the characters :)

Also i saw you won Stanlet Parable .... PLAY IT !
Its a masterpiece :)

If you like 2d Indie Platformers I would suggest:

Full Bore
Fly'n

I have reviews for those if you want a personal opinion.

Other than that I can also recommand Guacamelee and Dust: An Elysean Tale

There are more but this shall be enough for the moment.

Did you say open world?

JUST CAUSE 2

Still one of the best games of all time. Tons of extra fun with easy-to-use mods as well

I like (in no particular order):

1. The Monkey Island game series by Lucas Arts.
2. Life is Strange
3. TellTale Games gems: The Wolf Among Us, Tales From The Borderlands even though I've never played any BL games, etc.
4. Botanicula and other games by Amanita Design
5. Guns of Icarus Online
6. Kingdoms of Amalur: Reckoning
7. Thomas Was Alone
8. Dust: An Elysian Tail

And the list goes on and on... of course it's a bit hard for me to recommend you anything since we like different things, but there goes!
Ps: Also, as you can tell, I'm more of a casual gamer, so.

Steven Universe ftw! <3

Story-Dribben Shooters, like Half-Life or BioShock

Deus Ex 1 and Human Revolution. Although they are more like stealth games with the option to shoot…
Mass Effect 2-3 or Alpha Protocol. Classified as RPGs for reasons of people not knowing what a cRPG is, but they are great story-heavy console third person shooters.

Point-n-Click, like Little Inferno or Grim Fandango

Pretty much any Daedelic title. Deponia is a good start, although my personal recommendations are Night of the Rabbit and Chains of Satinav + Memoria. CoS is quite bad, but needed for Memoria which is really, really, really good once you realise what the story was actually about.
Any TellTale game before The Walking Dead. Since then they make movies, but for years they made the greatest episodic point 'n' clicks. I absolutely recommend the first two seasons of Sam & Max and the Back to the Future games.

Action, like the Batman Arkham series

Remember Me. Also classifies for story driven. Game has low ratings because the publisher essentially butchered it (what else to expect for Capcom, huh?), but even this product is easily on par with the Batman games… and in some respects beats them.
The two TransFormers Cybertron games. Made by people who actually know their TF lore (ahemnotlikeacertainmoviedirectorahem) with a really fast-paced and interesting combat mechanic and good level design.

I'll recommend Shadowrun Returns. An story-driven-rpg-future-steampunk-like indie game.
Dead Man's Switch Campaign is a great story, and there are custom maps too from steam workshop.

deus ex and dx:hr.

machinarium is pretty nice.

Batman Arkham Asylum of City.
Action: Darksiders I & II.
If you wanna laugh play Battleblock theater.
Open World: Sleeping Dogs or GTA any of them.

I have a group where I recommend most of the games I've enjoyed on Steam. You can check out what I've recommended here.

Tomb Raider is great fun

For point and clicks/story driven, definitely Life Is Strange! There is one more episode to come so you can catch up in time :D