Use the Zotero bookmarklet when:
You'll probably use the bookmarklet to supplement Zotero for Firefox or Zotero Standalone rather than replace them, since working in your online Zotero account doesn't offer as much functionality as the full Zotero program.
If you're using Internet Explorer or a mobile browser, such as Safari for iPad or the Android browser, you can still save items to your Zotero library using a bookmarket. A bookmarket is a special link that saves items directly to your Zotero account on the web (if you use Pinterest, the "Pin It" button is an example of a bookmarklet).
First, set up a Zotero account. Then visit the Zotero Bookmarklet page for installation instructions. You won't have to install any software -- just the bookmarklet!
Many thanks to Jason Puckett for creating the Georgia State University Library's Zotero guide, which is the source of much of this guide's content!