How to provide a translation for a string used within t()? On my site I use a module which creates text using t('text from module'). How can I tell drupal that I have a translation for that, and of course, that my translated text is being displayed?
First, make sure you install the Localization update module.  It is the key to handling anything that has to do with translation.  Enable automatic translation updates.

If the module is an "official" module with a stable release. the Localization update module will pull down translations from localize.drupal.org - provided they exist.  If they don't exist, you should provide them (read about how you can contribute on localize.drupal.org) - and the Localization update module will install them for you within 24 hours (if you've enabled automatic updates).

If the module does not have a stable release, there will not be anything on localize.drupal.org.  In that case, you must add local translations using the built-in translation interface.