For example, you can use a digital recording program, like Audacity, to record radio online a podcast. You'll then want to encode it into a format that can be streamed online. You can then use an app like Icecast to broadcast your show.
You'll also want to check out specialist apps like Spotify, Pandora, and Google Play Music. These services will allow you to search for specific songs and artists. However, they are not always free to stream. This is because royalties have to be paid to the artists who perform the music.