PVR stands for personal video recorder, and DVR stands for digital video recorder. Both technologies refer to devices that can record and playback digital video content.
Before the arrival of PVRs, analog VCRs were used for recording video on to a tape, which then could be played back. Many VCRs also supported timed recording, where one could set a time to start and stop record, or even set up recurring recordings. Analog recording quality was not that good, and recording time on a tape is typically limited to a few hours. The tape is a sequential access device (as opposed to random access), so jumping to a point in the tape always has to wind through the rest of the tape.
PVRs/DVRs changed all this. It is possible to record many hours (over 100 hours of recording is common now), all of it in full digital quality. If connected to a digital transmission (cable or satellite), the PVR can record the content with practically no loss in quality. Most PVRs use some sort of program guide, which serves in channel navigation as well as to set timed recordings. Trick plays allow quick fast forwards and rewinds, getting the viewer to any point in the recording almost instantly. An interesting feature supported by most PVRs ia that they allow to pause, rewind and fast forward live program. This is achieved by continously recording the program so that if the viewer pauses, recording continues. Note that fast forwarding of live program only works if the viewing is behind the live program.
There are multiple options for buying and setting up a PVR. Living room VCR-like boxes such as TiVO and ReplayTV are the most obvious solutions. A cheaper setup may be done on a PC using a capture card and PVR software. It is also possible to build dedicated media PCs with dedicated hardware for video/audio interfaces.