You are able to access/manage all citations online when logged on to zotero.org, but some functionality and special features, like folder importing/exporting, are only available when using Zotero Standalone.
The browser connector will allow you to save items to your Zotero Library from your browser. The mobile version of zotero.org allows you to access and edit your Zotero library on your tablet or mobile phone. See the Using Zotero with a Tablet/Mobile Device tab of this guide for info about additional steps.
After you download the software and browser connector, register for a free account.
This will allow for:
If you're regularly using more than one computer or device in your research, Zotero's sync feature can keep your library up to date on all of them. Zotero can store a copy of your library on the Zotero.org server and check it for updates whenever you open your library on a different computer.
These should match the list of citations from your Zotero Standalone. The initial syncing should happen within a few minutes, if you do not see your citations included try refreshing your browser cache and reloading the page.