Mac
Open the .dmg you downloaded and drag Zotero to the Applications folder. You can then run Zotero from Spotlight, Launchpad, or the Applications folder and add it to your Dock like any other program.
After installing Zotero, you can eject and delete the .dmg file.
Windows
Run the setup program you downloaded.
Linux
Download the tarball, extract the contents and run zotero
from that directory to start Zotero.
Chromebook
To set up Zotero on a Chromebook, see Installing on a Chromebook.
Integration with Google Docs
Hot to add a Zotero toolbar and menu to the Google Docs interface:
Zotero Google Docs integration