The Cite pane has two tabs: “Styles” and “Word Processors”.
The Style Manager displays the currently installed citation styles and the date they were last updated. You can download additional styles directly from the Zotero Style Repository by clicking the “Get additional styles…” link. You can also install a local Citation Style Language (CSL) style file by clicking the “+” button and locating the style file on your computer. To delete a style, select the style and click the “-” button.
If you aren't sure what style you need, you can Search by Example to find a style. Note that this tool requires you to format the reference data shown on the page, not just any example reference.
Zotero's word processor plugins integrate Zotero into either Microsoft Word or LibreOffice. Zotero will install plugins into Word and LibreOffice automatically when you install Zotero. You can re-install the word processor plugins from this pane. Reinstalling the LibreOffice plugin may be helpful if upgrading LibreOffice causes the path to the LibreOffice program files to change.
If one of the Install buttons is disabled, check that the respective word processor extension is installed and enabled in Zotero Add-ons window (click “Tools → Add-ons”).