When Google said Hangouts would be available on all platforms it wasn't exaggerating. The new messaging service is available on mobile devices, of course, on Android and iOS as native apps. It's also available on the web, in Gmail and in Google+.
However, if you're on your computer and don't want to keep a Gmail tab open all the time, there is an alternative, a Chrome Hangouts app.
The Chrome version of Hangouts is based on the new APIs. It's a new packaged app, meaning it runs in its own window independent of Chrome.
For users, Hangouts for Chrome will look and feel pretty much like a regular instant messaging application. There's a system tray icon for quick access, a contacts list, and chat windows for the individual conversations.
There are some issues, and they have more to do with the packaged app APIs than with the Hangouts app itself, which is why the technology isn't yet advertised, even if it's supported in Chrome. That's also why you won't find the Hangouts app in the Chrome Web Store by searching or browsing, you have to go to the direct link.