First, you need to download the latest release from the front page. Everything you need to start using Growl, including some extras and scripts that we will detail later, are included in a disk image.
To Install the application, open the disk image, and double click in the Growl.prefPane icon.

Just like many services that are included in Mac OS X, Growl is designed to work in the background. You will not find a Growl window, nor an icon on your dock.
Open the System Preferences and click on the Growl Preference pane, located near the bottom of the window. Inside, click on the Start Growl button. Optionally check the Start on Login checkbox; if this is not checked, you will need to manually start Growl in here, each time you start your mac.

If you click on the Applications tab, it may be empty. You will need to register some applications first.
Growl is what we like to refer to as a tool for other applications. Now, we don't want to include a list of all of the possible applications in the default installation, because you probably will not be using them all. We also don't know if you would want to even use the applications we have made as examples. So with the default installation of Growl, the applications tab is meant to be empty.
Have no fear. Soon it should fill up with whatever Growl enabled applications that you may have (see here for a list).
When a Growl enabled application tries to talk to Growl, it does something called Registration. It registers with Growl so that Growl knows what notifications it may send out (so that Growl can list them in its preferences). This is true for scripts as well, anything from AppleScript to Tcl will register.
For the most part, all you will have to do is just launch the application. There are some exceptions. Some applications use filters or rules or whatever was chosen to allow their users control over the whole process within their application as well. If the application you want to use with Growl does not show up in the Applications tab after launching it, look in preferences. If you still don't see it, you can email us at the discuss list and we can look into the problem.
Once the application has a registration in the Applications tab, you can control it. You can control all Growl based notifications through this one central location. You can also choose how each notification will appear. One notification can use the Bubbles display, another can use the Music Video display, and a third can use the Bezel display. You can also choose which applications will stay on the screen until clicked, and which ones will not.
Other preferences, are detailed in Exploring Preferences.