DVR is a piece of hardware. It stores digital signals like a VCR (no more tapes!).
TiVo (yes, you MUST capatalize the T and V!) is a service. Used in conjunction with a DVR it gives you name based recording options. You may create season passes witch will record whatever show you specify (new only or new and reruns). Wishlists search for titles, actors, etc...
Most (if not all) HD DVRs use HDCP to prevent you from recording HD content for copyright reasons.