It's too bad you didn't mention your actual platform (iOS, Android, Windows Phone 7, Blackberry, etc.).
On Android, however, I can recommend two different apps:
Audiogalaxy will stream music straight from your PC to your phone. Now, mind you, I haven't used it in about a year, but it worked well, was very stable, quality was top notch and you could make and save playlists. Album art would show up fine. Very slick for something that's completely free. The only catch is you have to install the software on your PC and leave your PC on at all times.
The reason I no longer use Audiogalaxy: the hard drive that housed my music has gone faulty and no longer streams without crashing the whole computer. That's okay, though, because before it fucked up on me, I had already uploaded all my music to Google Music, which streams to your phone from Google's servers, and thus, you do not need to leave your PC on at all times. Even better, it's also completely free. Google Music will host up to 20,000 tracks (yes, they measure your music by tracks, not megabytes!). It allows you to manage playlists, thumbs up/down tracks, and has auto-playlists. I also highly recommend this service.
Both services allow you to log in to the website from any internet-connected computer and play music right in the browser.
If you don't have android, then I have no idea what will serve you best. Google Music may still be available as a web-app on iOS.
