Zotero is a program that allows scholars quick and easy ways to collect, organize, and cite scholarship. The program itself contains robust tools to manage and organize lists of sources, read and annotate them, and then generate bibliographies and citations with the click of just a few buttons. In addition to the desktop program, though, Zotero also uses a tool called a "browser connector" to link sources found online to the local program.
All forms of Zotero require a Zotero account to sync. Accounts are free, and users are allowed 300mb of storage, which allows for the cloud backup or "syncing" of Zotero libraries. Increased storage amounts are available for purchase, if you require more, though you always have as much room to save your Zotero libraries as your computer has local disk space.
Zotero is an application installed on your computer which manages your bibliographic library locally, and syncs data to the Zotero storage cloud using your Zotero account.
Zotero Connector connects Zotero to most modern internet browsers, allowing for the quick and easy capture of your citations from websites, library catalogs, and databases.
Both Zotero and the Zotero Connector should be updated automatically by default. However, you can always manually install a new version of Zotero over your existing version without losing any data.
The first thing you must do to use Zotero is register for an account. The free account option comes with 300mb of storage, though options for upgrading your storage amount are available.
Once you have an account, there are several ways to use Zotero:
Zotero: Zotero (currently on version 6) is a self-contained desktop application where you can view and manage your bibliographic citation library. Zotero regularly syncs file storage to Zotero's cloud servers automatically (when connected to the internet). You can manually add citations and attachments to your Zotero library, or you can import web-based sources by using the web browser-based Zotero Connector.
Zotero Connector: The Zotero Connector installs into your internet browser of choice and links the Zotero desktop application to your Zotero library. Once installed, the Zotero Connector toolbar button will save your browser's current page (and certain file attachment types) to the Zotero collection you've selected in the Zotero desktop application, as a bibliographic entry. In order to save an entry from the Connector, you must also have Zotero open on your desktop.
Zotero Bookmarklet: If you do not wish to (or cannot) install browser add-ons or software on your computer, you still have the option of using the Zotero Bookmarklet from (almost) any browser, including mobile/tablet browsers. In this case, clicking on the bookmarklet (or using the link https://www.zotero.org/save) will save the items directly to your Zotero cloud-based online library at zotero.org. However, all saved items go to one folder (you'll have to organize them later), and not all websites can be scraped using the bookmarklet. This option does not allow you to use Zotero's word processor plugin.
Zotero.org web library: Zotero also allows you to browse your library through their website. Here you can find all the sources and attachments you've saved, and can organize your collection directly through your browser. Though you can manually add and organize entries through this method, you will not be able to automatically grab web resources this way.