Google Music: https://music.google.com
Google Music Store: https://market.android.com/music
In this video I give a hands on tour of the new Google Music along with the Google Music Store inside the Android Market. With Google Music you can upload up to 20,000 songs for free as well as get quite a bit of free music. Google Music is available in your browser (desktop and mobile) as well as a dedicated Android app.
You can also buy music with the new Google Music Store. The selection is reasonable but not perfect just yet and pricing is very similar to iTunes.