Is it possible to run a module written for drupal 7 in drupal 6? In another question I was searching for a module that provides a simple shopping cart and checkout process (Is there a simple shopping cart and checkout process module?). 

I have found this simple module called Basic Card.

I am running Drupal 6.x while this module is written only for Drupal 7.x. 

Is it possible to run a Drupal 7.x module on a Drupal 6.x installation?
No, it's not generally possible without back-porting the code.

The API is very different between Drupal 6 and 7. A few things have remained the same, so it's theoretically possible that a module for one version might work as-is in the other version, but generally speaking, it's not possible.

For a proper idea of the task ahead of you, take a look at Converting 6.x modules to 7.x. Not a trivial task, unfortunately.

I just had a quick look and the specific module you've mentioned definitely won't work in Drupal 6, it would need to be re-written.