The online version of Zotero is quite robust. In order to use it you must create a Zotero account and sync your Zotero library to Zotero.org. This is also where you can create and manage Zotero Groups (although the actual moving around of records between your Zotero library and a group library happens in your Zotero library on your desk/laptop).

Your Zotero library on Zotero.org lets you add records via cut and paste, edit records (including adding notes and uploading attachments), and cite works using about 8 commonly used citation styles. If you use the Zotero bookmarklet, anything you add is automatically added to your Zotero.org library, and then synced to your desk/laptop Zotero library. 

If you just sync your Zotero record (including notes) it will not cost you anything. If you choose to also sync attachments (i.e. PDFs and webpage screenshots) you get 300MB for free, then there is a charge. I pay $20.00 /yer for 2GB - this pays for my storage and helps to support Zotero (its probably the most valuable $20 I spend in a year ;-) Syncing attachments means you have access to your article PDFs, etc. wherever you are, as long as you have an Internet connection. Note: there are WebDav and DropBox/GDrive/OneDrive/etc.storage options for saving attached files as well, although they don't provide the full experience you get if you use Zotero's own storage option)

Note: mobile device apps such as PaperShip and Zotfile can access your content in Zotero.org and provide added value. For more information on this, see the Zotero on Mobile Devices tab on this guide.