This method works with any word processor (or anywhere you can paste text).
Zotero offers word processing plugins for Word and OpenOffice. 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. Mousing over the toolbar will pop up each button's function if they're not clearly labeled.