In order to install Zotero, you first must have Firefox as your web browser (if you want to use all of the Zotero features).
Once you have Firefox, go to the Zotero webpage and click on Download. You will be prompted to set up a new user account.
Once you have your account setup, you can choose which of the Zotero products you wish to install and use.
Zotero for Firefox is needed to use most features.
If you want to be able to generate bibliographies, then you will also need the Word Processor extension.