Zotero Standalone runs as a separate application and plugs into your choice of browser. Standalone has connectors that work with Safari and Chrome browsers as well as Firefox.
You can not have Zotero in Firefox and Standalone open at the same time. You must have Zotero Standalone open to use the Chrome and Safari capture icons. (If you don't you will just be redirected to Zotero's website.)
You do not have to install the word processor plugins with standalone. The plugins are bundled with the software download.
Getting Zotero: Installing Firefox extension.
To install the Zotero extension in your Firefox browser, visit http://www.zotero.org and click the red download button.
If your pop-up blocker is enabled, you may see a yellow bar at the top of your browser telling you that Firefox has prevented the site from installing software. Click Allow
Word Processor Plug-in.
An additional word processor plug-in is needed to insert and format citations in Microsoft Word or Open Office documents when using the Firefox extension. The plug-in you install depends on your operating system and word processing program. To find and install the right plug-in and extensions for your word processor, visit http://www.zotero.org/support/word_processor_plugin_installation
Once you have successfully installed the plug-in, you will see the Zotero citation tools right in your word processor.
Here is an example of what the Zotero plug-in looks like in Word 2007.