If the source allows it, yes. Many do not, so you'd have to record them using a speaker and microphone, as you would any other standard radio broadcast.

Many internet programs are available as a podcast, which is basically an MP3 file you can download and play back from your computer or MP3 player (does not need to be an ipod). For "manual" recording, Audacity is a very good freeware, open-source program. There are some programs that allow scheduled recording.