not sure where the term comes from, maybe back from analog TV days. But basically the output of an RF diode always has some series resistance and some shunt capacitance. So even though the diode itself might be quick enough to rectify GHz rate signals, the DC output can not change very quickly because of that RC time constant (and what is loading the diode, like load resistance/capacitance). A big part of the series resistance is the diodes own "dynamic resistance", which you can measure by the slope of the I-V curve. Often the rising edge of a detector output is quick, but the falling edge (where the diode turns off) is slow unless the diode is loaded with 50 ohms.
To determine the "video bandwidth" one AM modulates an RF signal, applies it to the detector, and sweeps the modulation rate from perhaps 100 Hz to 1 MHz. At some point the sine wave video output drops by 3 dB, and that is what you call the "video bandwidth". Very useful when you have high data rate digital AM modulation....if the video bandwidth is not wide enough, the modulation bits average out to a flat DC voltage and can no longer be detected. i.e. you diode envelope dector becomes a DC power rectifier instead.