GAMES

To the Moon: Story 10, game 3, requirements 1.

Gothic 2.

i'd say bioshock(the first one). amazing story, great gameplay and runs on most computers

Telltale games.

The Walking Dead

Minesweeper

The Wolf Among Us. Half Life series(with Black Mesa).

Fallout 1-2, Planescape: Torment, Sanitarium.

Bad Rats

What kind of games do you like? FPS, point&click adventures, action, indie etc. ?

Also, Fahrenheit/Indigo Prophecy, Portal, Portal 2, The Stanley Parable, The Walking Dead, The Wolf Among Us.These have low requirements and great stories.

Mass Effect

Spec Ops: The Line.

Thomas Was Alone (minimalistic style, but yet a nice story)

fortix 2

Dead Space

skyrim :-)

The Walking Dead series is pretty awesome, and has relatively low system requirements.

Few games make you care so much about the fate of a collection of pixels ;__;

Minecraft

Prey (2007)
A very underrated game with a badass/f*cked up story, innovative epic gameplay and superb graphics for that time, runs on a calculator.

You can't find it in the Steam store though. Its gift is a collectible now.

The Swapper

The Walking Dead and The Wolf Among Us

Minimum:
    OS: Windows XP Service Pack 3
    Processor: Core 2 Duo 2GHz or equivalent
    Memory: 3 GB RAM
    Graphics: ATI or NVidia card w/ 512 MB RAM
    DirectX: Version 9.0c
    Hard Drive: 2 GB available space
    Sound Card: Direct X 9.0c sound device
    Additional Notes: Not recommended for Intel integrated graphics 

Recommended:
    OS: Windows 7
    Processor: Core 2 Duo 2.3 Ghz or equivalent
    Memory: 4 GB RAM
    Graphics: ATI or NVidia card w/ 1024 MB RAM
    DirectX: Version 9.0c
    Hard Drive: 2 GB available space

Bad Rats, Story 11/10

System Shock 2 (Horror FPS), Deus Ex (Sci-fi FPS/RPG), Planescape Torment (RPG)

I'd probably go for the old RPGs. You know Baldur's Gate, Neverwinter nights, Planescape Torment, etc.