Check to see if your DVR has a
firewire (IEEE 1394) port. I know that cable providers are required by law to provide cable boxes with firewire ports upon request. Cable boxes usually have firewire, but DVRs don't always. If yours does, you'll just need a firewire cable to capture your TV shows in HD. It will probably be a 6 pin connector that looks like
this (but could be a different size and shape).
There are free utilities for both windows and mac that capture video over firewire and this is by far the easiest way to get video off your DVR. This will transfer files in real time.
The next best option is to see if you can network into your DVR and transfer files over ethernet to your home network or directly to a computer. This will probably be something you'll have to lookup online with your specific type of DVR. I use this method to transfer files from TiVo, but I don't know if you can log on to every DVR over LAN. Depending on how your DVR is set up to network, this could be faster than real time.
If you have a digital camcorder, check to see if it has AV/DV passthrough. This essentially allows your camcorder to function as a capture device for analog video (Your HD shows will be converted to analog by your DVR). This function is actually available on a lot of inexpensive (and somewhat old) camcorders. You'll basically take the 1/8 to red, yellow, white composite RCA cord that is normally used to view your camcorder on the TV and plug the red yellow white into the output of the DVR and the 1/8 into the camcorder. Then plug the camcorder into your computer using (probably) firewire (or maybe USB, depending on your camcorder). Turn on the AV/DV passthrough mode and capture the video on your computer like you normally would from your camcorder. (The end result of this method is going not going to be HD, but if you're capturing from an HD source, 480 anamorphic widescreen video does not look bad at all.) This will transfer files in real time.
If none of the above options are available, you'll probably have to look into getting a capture box such as one from Hauppauge, Blackmagic, etc.
An alternate route would be to ditch the DVR, get a HTPC with a cable card tuner, and use that as your PVR. IMO, this is really the best option, but the most expensive to purchase. (Would probably cost minimum $1000 for a good HTPC and cable card tuner.) However, it could save you money in the long run without the monthly cost for a DVR.