In days of yore, machine maker put 'stops' onto control knobs; a spring and plunger usually that 'latched' into a hole on the knob's shaft to make it 'stop' turning at pre-determined positions.
Camera makers found this rather convenient for making aperture and shutter speed knobs 'stop' at pre-determined settings that were convenient to photographers, and represented a halving or doubling of exposure...
Eg: a shutter speed of 1/250th is twice as long, so twice as much light as a shutter speed of 1/125th; a shutter speed of 1/500th is half as long, so half as much light; hence 1/500th is 'one-stop' less exposure, 1/125th is 'one-stop' more exposure.... F-Numbers are a bit peculiar because they are actually a ratio of the aperture diameter to focal length, before they set the 'stops' to match shutter-speeds, halving or doubling exposure, which accounts for the some-what peculiar mathematical progression, B-U-T thanks to 'stops' we don't really need worry about it.
FILTERS... dim the light let into the camera.. a Neutral-Density filter is probably the simplest to explain; it just dims the light entering the lens across the whole scene and to keep things 'easy' they are usually sold, and marked with the 'stop value' of how much they reduce light levels..
Probably the most common and cliched use of ND filters is to 'milk' waterfalls.... Yuk.... but the idea is you want to use a relatively low shutter speed in order that the moving water 'streaks' across the frame during the exposure... So point camera at waterfall, drop the ISO as low as you can, and shut the aperture as low as you can go... and look at the shutter speed..... on a good clear sunny day, f-16 sunny says a shutter speed 1/ISO setting, and with many cameras lowest ISO being 100, that would probably only get you down to a shutter of perhaps 1/50th at f22.. which is still a relatively 'fast' shutter and not likely to do much water-fall milking... so you probably want to.... first....use a tripod!.. but then get the shutter down to something in the order of 1/25th would be one stop, 1/12th two, 1/6th three, and so on.. so, to get the shutter sped down that low, you would want a one, two or three stop ND filter... the number of 'stops' your preferred settings differ fro the ones suggested by the meter....
That's the 'principle'.... BUT in practice, you can chuck plenty of the simplified maths out the window.... if you are milking a waterfall, the meter is taking a 'instantaneous' light level reading; what is making the 'milk' in your picture is the highlights smearing down the frame... that light-level reading taken instantaneously then is ONLY good for a something very closely approximating the 'instant' the meter takes its reading, ie a very very short shutter... the waterfall, your 'subject' is predominantly those high-lights, which are traveling down the frame, exposing a highlight on a different bit of the sensor as they move, so if you shoot 'strictly' to the numbers, your water fall can come out 'very' milky, and a lot more 'over' exposed.. as one hihlight moves over a portion of sensor already exposed if not saturated by the last; which where experience and judgement starts to come in, and how much to adjust the 'exposure' so that the highlights don't utterly 'blow' a large region of the frame, and you get 'some' detail in the water, not just a milky mess.. and you start considering alternative metering modes or methods, like spot metering, or incident rather than reflected readings, in order to work from a better base-line, but probably still 'adjusting' and more, the lower the shutter-speed, compensating for the exposure occurring over an extended period, and the exposure metering being taken instantaneously... make sense?
In other circumstances... you might use an ND filter the other way about; in order to use wider apertures for 'selective' (OK, shallow!) focus, in bright day-light, when even modern cameras super-fast 1/4000th ish fastest shutters could still be erring towards over exposure, if it's an f-16 sunny day, and you want to use and f2 or lower aperture, which would beg a shutter speed @ISO100, of 1/2000th or more...
In the middle, you have the perms where you may, say, want to use a slower shutter to get a bit or movement in say trees or grass, but still use a wider aperture to chuck background OoF.. where circumstance dependent, the maths may more easily translate..... B-U-T most common is probably the water-falls, where it definitely doesn't.
'Variable' ND filters... worth a mention... imagine a polariser as a grill for light.. lets light vibrating up and down to pass through the gaps, anything vibrating side to side, hits the bars and gets blocked. Put two one infront of the other.. when the grills are in line, what gets through the first, will go through the second.. twist the second filter in relation to the first, some of what gets through the first, may get blocked by the second, until when the two are at 90 degrees to each other, they effectively close the holes, and nothing gets through..... that's how they work, BUT because they are variable, they cant be conveniently marked with a 'stop' value of light reduction... a single polariser cant anyway, because its blocking effect depends on what 'proportion' of light is vibrating along one or other axis, and its orientation to that light.. so you have to work to the meter with both plain polarisers and variable ND flters, and measure its effect on the spot, which may make them more trifficult to work with for 'big-stoppa' milky waterfall effects.
But thee's your answer; they know that a stop is one setting change. so if they want the same 'exposure' but wish to use so many stops more aperture, or so many stops faster shutter, then they need that many 'stops' of filter to get it all to balance... hence hinges on knowing that bit of 'jargon' of what a 'stop' is, really.
