As Drupal develops with each release it becomes necessary to update modules to take advantage of new features and stay functional with Drupal's API.
The Coder project provides automated tools to assist you with the process of updating a module. From Drupal 5.x on, the Coder Review module can assist with a module upgrade by reviewing a module's code for things that need to be updated between versions. Beginning with Drupal 7.x, the Coder Upgrade module can assist with a module upgrade by changing a module's code to correspond to a new API version. While not perfect, these two modules can greatly assist you with the process of updating a module.
Lullabot features a useful video tutorial on how to upgrade your Drupal module to a new version.