To open PluginProcess, you need a Java runtime. Would you like to install one now?A search on this string will produce many hits, most of which are terribly misleading. So I'll add another one, which is probably also misleading.
This message comes because a "Safari" or "Internet" Plug-In ("plugin) is trying to run, and that plugin requires Java. Java used to be a part of every Mac, but Java in general, and Java in the web browser, have been an endless security nightmare. Apple, wisely, disabled Java browser plug-ins in a security update and stopped including Java in Lion and Mountain Lion. If you have the plug-in disabled in Safari you could reenable it (but don't), or if you don't have Java at all you could accept the download suggestion and install it (but don't).
Don't do that, however. No modern plugin should require Java on a Mac. This message is probably a sign that you're unwittingly running an ancient plugin that you don't want. It's particularly suspicious that you're seeing "PluginProcess" rather than the name of a specific app.
I don't know of any easy way in Safari to identify Plugins, much less the problematic ones that probably aren't running properly. (Extensions are a different beast). To eliminate this ugly prompt I had to navigate through several Library folders, deleting all I saw. Of course this will also delete desired plugins (if any exist) but you can reinstall those.
Delete everything from these folders (on the troubled account and globally) then restart and recheck:
User account: Library\Internet Plug-Ins
Root: Library\Internet Plug-Ins
I saw some gnarly old things in there.