In the UK you can use Click to Play (streaming) for both radio and television programmes on the BBC iPlayer website, and radio programmes have a pop-out player for you to enjoy listening in the background.
If your computer supports BBC iPlayer Desktop you can also download programmes to play up to 30 days after broadcast. Both regular and HD are available to download.
Outside the UK you can use RealPlayer or Windows Media Player to play programmes.
You can also get advice if you are in the UK but told that you're not.