You can download Zotero on the Zotero download page. Be sure to also install a Zotero Connector for your browser.
Run the setup program you downloaded.
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.
Download the tarball, extract the contents and run
zotero from that directory to start Zotero.
For Ubuntu, the tarball includes a .desktop file that can be used to add Zotero to the launcher. Move the extracted directory to a location of your choice (e.g.,
/opt/zotero), run the
set_launcher_icon script from a terminal to update the .desktop file for that location, and symlink
ln -s /opt/zotero/zotero.desktop ~/.local/share/applications/zotero.desktop). Zotero should then appear in your launcher.
Alternatively, Linux users may wish to use a package for their distro rather than using the tarball directly. Note that such packages are built by third parties, and Zotero can only provide support for the official tarball.