You need computer-based Zotero to use Web-based Zotero.org.  See the Zotero page to install and use.

Open up your computer-based Zotero panel, go to the gear icon and set Preferences for Sync.  Enter you Zotero.org username and password.  Be sure to erase this afterwards, if you're on a public or shared computer.

I recommend that you not check the box for "Sync automatically." 

When you want to transfer citations from your computer to your Zotero.org account, or transfer them back to another computer, click on the Sync icon on the upper right of the Zotero panel.  It's the green, clockwise arrow.