HDR are Hard Disc Recorders. These are any of a variety of set-top boxes that contain a computer hard drive. Programs can be recorded into an event queue but it even records movies and TV shows while you watch in real time. "Real time" means the HDR is constantly recording while you watch, so if you miss something you can rewind or pause. The biggest advantage of the HDR over VCR is video quality and convenience.
The HDR records in MPEG-2, the compression format used by satellite and digital cable. The audio quality of recordings on HDR isn't limited to two channels like a VCR. The HDR will record and play back just as it was transmitted to the receiver. This means not missing out on surround effects when 5.1 broadcasts are recorded and played back on a 5.1 capable HDR.
The most popular proprietary HDR brands include Tivo and ReplayTV. Each has their own menu system and features.
Fast approaching Tivo in popularity are the subscriber TV services line of HDR receivers. Digital Cable and satellite receivers can include HDRs that conveniently integrate with the subscriber service's menu system.
Using either the HDR brands or your subscriber TV service means convenient selection from an electronic program guide. There's nothing like going away on vacation and having all your favorite shows waiting for you when you get home.