The mobile version of the Zotero website (zotero.org) allows you to access and edit your Zotero library on your tablet or mobile phone.
The Zotero Bookmarklet allows you to save items from your mobile device's browser directly to your zotero.org account.
For most users, using the mobile site and booklet will be sufficient. If you do a lot of mobile research, there are also third party apps available.
PaperShip is available for iOS, and can be used with Zotero and other citation managers.