Google Music Store is available in the Android Market (both the web interface and a new version of the mobile app that will be released soon). A surprising feature is the integration with Google+: "if you use Google+ to share a song or album with someone either privately or through a circle, the person who receives the share will get one free full play of the song or album. If you do a Public share, people in your circles will get one free full play of the shared song or album. Everyone else who sees the share will get a preview."
Google offers a lot of exclusive songs and albums, but you can't download the free songs from the store without associating a US credit card to Google Checkout.
Google Music will store the songs you've uploaded or bought and now you can also download them. For example, in the web interface, click the arrow icon next to a song and select "Save to computer". The catch is that "you can only download each purchased track from the web 2 times". To download all the songs you've purchased from Google Music, use the Music Manager app.
"Google Music is about discovering, purchasing, sharing and enjoying digital music in new, innovative and personalized ways. Google Music helps you spend more time listening to your collection and less time managing it. We automatically sync your entire music library — both purchases and uploads — across all your devices so you don't have to worry about cables, file transfers or running out of storage space," informs the Google blog.