Google released a SDK for Google Drive apps and there are already 18 apps that use it. Users are required to install a Chrome app from the Chrome Web Store for each Google Drive app, but they don't have to use Chrome. Google's dev blog informs that "Drive apps are distributed from the Chrome Web Store and can be used with any modern browser." I've managed to install apps in Firefox and Opera, but not in IE9.
To integrate a third-party app with Google Drive, the app sends you to a page that asks for your permission. The authorization process fails if you load the page without installing the Chrome Web Store app. As Google says, "apps will not have any API access to files unless the app has been installed in Chrome Web Store." Chrome apps are only used to allow the third-party apps to use the Drive API, but they can be uninstalled immediately after they're installed and all the features will work in Chrome and any other browser.
At the moment, Drive apps don't integrate with Chrome, but that's likely to happen in the future. Chromebooks would be a lot more useful if you could sync files with Google Drive and open them using web apps.