Google Play Music lives on the web and as several mobile apps. But as you would expect from Google, there's not much in the way of desktop support. If you do want to air your tunes from a desktop application in Windows or Mac OS X, then there are a couple of third-party, open source options available: Here's how to find and use them.
First up is the imaginatively titled Google Play Music Desktop Player for Windows (hat tip to SlashGear). While it sticks to the basics, it does have one useful feature you can't find on the web: Last.fm integration. Sign in with your Google credentials, and you get pretty much the same interface as you see when you load up the app in your browser.
There's a new Desktop Settings entry in the menu — from here it's possible to tweak the look of the application, enable or disable desktop notifications, and configure the aforementioned Last.fm support. If you're a serious music scrobbler then installing this unofficial desktop program is worth it for this feature alone.
If you get your operating systems from Apple rather than Microsoft, take a look at Radiant Player (hat tip to our friends at Lifehacker). Again, it's basically just a wrapper for the web interface, but there is support for the media keys built into your Mac so you can control playback without having to reach for the mouse or trackpad every time.
Last.fm support is included here too, as well as custom styles, a mini player, and desktop notifications (check the Preferences entry on the Radiant Player menu). It's been around slightly longer than the Windows version, and as a result, it's a little more polished, but both applications are worth a look. You'll have one less browser tab to deal with at least.