First open Zotero. In the example below there is an empty Zotero library. To recover the lost data for you Zotero library go to the Zotero data directory by clicking on the actions button (gear icon) in the Zotero toolbar and selecting preferences:
Next, after the Zotero preferences window opens, select the Advanced tab in the preferences window:
Now select the button Show Data Directory:
The Zotero data directory will now be open:
In your Zotero data directory copy the storage folder and the zotero.sqlite file:
Paste the storage folder and the zotero.sqlite file into a temporary folder on your desktop:
Now go to your backed up Zotero directory which you have stored somewhere else on your computer or on a remote storage device:
Open your backup Zotero data directory and select the storage folder and the zotero.sqlite file and copy them:
Paste the storage folder and zotero_sqlite file that you just copied from your backup data directory into the current Zotero data directory. Click the Yes to all button when asked if you want to replace the current files:
Now reopen Firefox and open Zotero. Your Zotero library should be restored to the backed up version: