The Slingbox is a network device that allows you to stream the audio / video signal using the Internet. You need to connect the Slingbox at home and hook it up to your router (to reach the Internet) and also to the TV source (cable box, DVR or satellite receiver).
The following links will provide you with all the information you must know about the Slingbox.
Just keep in mind that you will not need the Internet service if you want to get the Slingbox up and running on your own network. Regarding the last two questions, it will all depends on your Mobile device capabilities.