Download and install Zotero from the Zotero website.
(Note: There's a different version for each operating system: Windows, Mac or Linux.)
Connectors allow your web browser to save citations to your Zotero library.
The Zotero downloads page should automatically detect your browser and give you a link.
Each browser has its own version of the Zotero connector, so if you use Chrome, Firefox, and Safari you'll need to install all three.
Register for a cloud-based Zotero account to sync/ back-up your work. This will also allow you to access your library from any internet enabled device.
The mobile version of Zotero allows you to access and edit your Zotero library on your tablet or mobile phone.
Download the Zotero Bookmarklet to save items from your mobile device's browser directly to your Zotero account. While there are no official apps for mobile devices, there are several third party apps.