Firstly, any DVR should work with just about any cable, satellite, or regular aerial hookup that you have. As long as you can get a signal into your TV, you should be able to get a signal into your DVR and record it.
One thing to consider, up front, is what you need it for and how much you want to spend. If you don’t record very much, and your main aim is to get rid of VCR tapes, then you might want to consider buying a DVD Recorder rather than a Digital Video Recorder.
A Digital Video Recorder, with a hard drive, is generally still considerably more expensive than a DVD Recorder, which doesn’t have a hard drive and records straight onto blank DVDs. A few months ago, we bought a DVD Recorder precisely because we were getting tired of having bulky tapes lying around all over the place. We don’t record very much stuff for permanent use; we mainly use it for time shifting. We use DVD-RWs, so that once we’ve watched a show, we can just erase the DVD and use it again. I got this one for $60, and so far it has worked fine.
DVRs with hard drives are generally a lot more expensive, although quite a few cable and satellite TV providers will give you one free with your subscription. TIVO boxes can also be had pretty cheap, but require a monthly fee. I’ll let someone more experienced talk more about DVRs. So far, a DVD recorder has been fine for my needs.