That is THE questions these days on the internet.
The short answer is: if the stream-provider does not want you to do so - you just can't or at least you are not allowed to do so.
But with many things on the internet, there is a way of doing it and in many cases it is ok to do so - if you want to keep a copy for personal use or if you want to download/record a stream for which the provider decided to not use special protection or DRM (digital rights management). In these cases you may download the content and save it for later viewing.
If the audio/video you want to save is on a streaming server or behind a meta-file it might be difficult to download it with your browser. For this case a lot of helper applications exist.
The Grumpy Editor's guide to audio stream grabbers is a good point to start your dive into this realm. all-streaming-media.com is the other source to read for further reference.
On LINUX mplayer or StreamRipper might be the tools of choice for you. On DOS/Windows you may choose between Teleport PRO, ASFRecorder and StreamRipper. But most tools break if a new version of streaming media container appears and grabbing a stream from the net quickly becomes a science of its own.
Not especially a streaming media problem is to download videos from the newly emerging video platforms like YouTube and Google Video. The problem here is that the actual video file is hidden behind a flash-player instance and locating the actual video file is sometimes nearly impossible. Here comes keepvid to the rescue, which helps you track down this URL. A great feature is that you can bookmark the service as a bookmarklet which helps you to trigger the service from any page with the click of a button.
BUT, on everything you do: keep in mind to stay on the legal side of things!