Monica, yes, the standalone TiVo (whether Series 1 or Series 2) would basically be positioned in the same place in the equipment chain as a VCR. The signal comes into the satellite receiver and goes out to the TiVo which records it. The output from the TiVo goes to the television, where you view it.
Do you have (or are you considering) DirecTV or Dish Network? If DirecTV is your choice, you'd prefer the DirecTV TiVo combo units, where the TiVo and satellite receiver are the same unit. They can use 2 feeds from the dish for the dual tuner, but they can be used with just one feed, using just one tuner.
If Dish Network is (or will be) your provider, take a look at their own PVR 501 as well as the standalone TiVo. The 501 can record without a monthly or lifetime fee, but its software isn't as fully featured as the TiVo. I have both a 501 and a standalone TiVo connected to a second Dish Network receiver, and I use them both differently because of their different features.
No matter which setup you choose, you won't regret it.
Chuck