As esdwa pointed out, don't mistake what content providers don't want you to do with what is illegal to do. "Fair use" provisions allow you to make your own copies of content you have purchased for the personal use of you and your family, and many areas of digital content, such as saving copies of streamed content, currently have few legal precedents to determine how fair use applies to them.
For example, many free tools exist to allow users to save YouTube videos locally, some of which download them directly, while others capture the videos as they are streamed normally in a browser. One argument in favour of the legality of such practices is that once content passes into the realm of your home network, it is yours to log and save as you please. If content providers do not wish you to have that ability, it is their responsibility to employ encryption techniques to prevent the end user from saving the content. Since doing that is expensive and most people don't know how to save streamed content in the first place, most content providers consequently don't bother to employ those protections.
The fact that most people don't know how to save streamed content does not, however, make it illegal for those who do have the knowledge, as options such as the libpcap/WinPcap packet capturing software are freely available to anyone who wants to use them. Content providers who encrypt their content also don't infringe on people's right to log packets; the difference is that the logs aren't useful, because their contents are encrypted. It's a principle similar to using encryption on a home WiFi network. The airwaves used by WiFi are free for public access, so people are allowed to transmit, receive, and store data sent over such networks. Nearby people can always spy on your wireless network traffic, so it is your responsibility to encrypt it to make their spying useless.
If the content you're trying to save is encrypted, then you would have to delve into the realm of DVD recorders that Rammitinski suggested. While fair use provisions for such things have yet to be well defined, it is a safe bet that employing such a capturing technique would at least be a violation of the content provider's ToS and be grounds for them to terminate your service. Whether the provider's terms comply with fair use practices is another question. Few people have thus far challenged the legality of many Internet-based content providers' practices, and as a result they are trying to be as restrictive as possible to destroy fair use concepts in the hope that they can boost their profits by giving people the illusion that they must pay for the same content over and over.
