There is no official Zotero app for a mobile device.
There are multiple third-party apps that work for either iOS or Android. Not every app will do everything that the Zotero program does. See the list of apps here.
To view your web library on a mobile device, you don't need to use any apps.
You will, however, need to have registered for a free Zotero.org account and synced your account to Zotero.org. See how to do this here.
Just browse to Zotero.org on your mobile device of choice and log into Zotero with your username and password. Click on "My Library" to see the items you have stored in your library.
The Zotero Bookmarklet is a bookmark you add to your browser to save the page you are reading to your Zotero library.
The process for installing and using the Zotero bookmarklet varies depending on the browser you are using.
Follow the directions here for the following mobile browsers: