Zotero offers word processing plugins for Word, Libre/OpenOffice, and Google Docs. The plugin adds a Zotero toolbar to your word processor that allows you to add citations to your document while you write.
To add a citation, click the first button ("Add/Edit Citation") on the toolbar. Search for the reference you want to cite and press Enter. Zotero will add the citation at your cursor.
At the end of your paper, click the "Add/Edit Bibliography" button. Your bibliography will appear, and new citations will be added automatically. Change bibliographic styles with the "Document Preferences" button.
The toolbar looks different on different operating systems and versions of Word. Hovering your mouse over the toolbar will pop up each button's function if they're not clearly labeled.
To create a quick bibliography:
Select the references or collections you want to include. Hold the control key and click to select multiple items. Right-click one of the selected items and choose Create Bibliography.
Choose the bibliographic style you want, select Copy to Clipboard, click OK and paste into your word processor.
This method works with any word processor (or anywhere you can paste text).
In Zotero, select the "Edit" menu, then select "Preferences."
Click the "Cite" icon, then the "Word Processors" tab. Check the "Use classic Add Citation dialog" box. Click OK.