Installing and starting up Growl for the first time
Growl 1.3 and later
Installing Growl is really easy as of Growl 1.3. Growl is now in the Mac App Store. Growl in the Mac App Store is located here.
Growl 1.2.2 and Below
Growl 1.2.2 and below are installed via something called a disk image. Think of a disk image as a CD, but without the actual physical CD. Below is an example of what Growl 1.2.2 looks like when the disk image is mounted. You would double click to launch the Growl.pkg to the left of the "Double-click to install" text.
Growl 1.2.2 and below will run as a background process which you can configure from System Preferences in the Growl preferences. You can get Growl 1.2.2 and other older versions from the downloads page.