Can I use tivo with a digital converter box?

First of all, I'm a little confused about what exactly Tivo is and does. I'd like to get rid of my cable and just use a digital converter box. I'm looking into Tivo so I could still record tv shows I watch regularly. Can Tivo record shows that are on channels I would no longer receive, such as MTV or ABC Family? I'd like to keep up with some tv shows I watch, and I wasn't sure if my only option is to watch episodes online. I see that you have to pay monthly for Tivo, so I'm not totally sure what the service actually is, and I can't find this answer on their web page. Any info is greatly appreciated!