Notifications are a way for your applications to provide you with new information, without you having to switch from the application you're already in.
Here are just a few examples:
Safari has finished downloading a file (GrowlSafari)
You've received new email (GrowlMail or Thunderbird)
You've received an instant message (Adium, or iChat with Chax)
iTunes has started playing a song (GrowlTunes)
Your laptop battery is low, or has finished charging (HardwareGrowler)
Growl handles the job of displaying these notifications, so that applications don't have to all do it themselves.