roll-off
Past the cutoff, a filter does not slam shut; it fades the signal out gradually. Roll-off describes how steeply that fade happens, how fast the output drops as you move further into the stopband. A gentle roll-off lets unwanted frequencies linger; a steep roll-off cuts them off quickly. It is the slope of the wall, not the height of it.
Roll-off is measured in dB per octave (each doubling of frequency) or dB per decade (each tenfold). A first-order filter rolls off at 6 dB per octave, equivalently 20 dB per decade: double the frequency and the output halves. Each extra order adds another 6 dB per octave, so a second-order filter gives 12 dB per octave, a third-order 18, a fourth-order 24, and so on. The simple rule is roll-off equals 6 dB per octave times the filter order, or 20 dB per decade times the order. Far from cutoff this slope is what determines how well the stopband is suppressed.
You pick the roll-off to meet your stopband spec: if junk must be 60 dB down only one octave above cutoff, no first-order RC will do, you need a high order. The honest caveat is that steeper roll-off is never free. Raising the order means more components, and the sharper types (Chebyshev) buy steepness with passband ripple, while flat or gentle types (Butterworth, Bessel) trade away some steepness. Near the cutoff the actual curve also depends on the filter's Q, so the clean per-octave slope only holds well into the stopband.
A first-order low-pass with a 1 kHz cutoff is down about 6 dB at 2 kHz, 12 dB at 4 kHz, 18 dB at 8 kHz, one octave at a time. A fourth-order filter (24 dB per octave) is already 24 dB down at 2 kHz, four times steeper, but needs four times the sections.
Each filter order adds 6 dB per octave; steeper roll-off costs more sections and other tradeoffs.
The neat 6 dB per octave per order is the asymptotic slope, valid well past cutoff. Right around the cutoff the real curve depends on the filter's Q and approximation type, and a higher-Q section can even peak before it falls.