Reciprocity is a pretty simple idea. If you double the amount of light through the lens (by changing the f-stop), you can keep the same exposure simply by cutting the time the shutter is open in half.
As an example, if f8 at 1/125 of a second is correct, then so is f5.6 at 1/250, so is f4 at 1/500 and so is f2.8 at 1/1000.
Likewise, if you cut the amount of light through the lens in half, but double the time the shutter is open, f11 at 1/60 and f16 at 1/30 then you have the same exposure as the previous example.
So, shutter speed and lens opening are inversely related – reciprocal.
Back in the film days, you had to worry about ‘reciprocity failure’. This means that, although the relationship between f-stop and shutter speed is linear, the sensitivity of the film emulsion is not. Typically, this only becames noticeable with long exposures (>1 second, for example). The package insert in the film would describe a correction for the exposure when using long exposures. This is not a problem with digital sensors, but digital sensors have their own problem with long exposures – noise.
Anyway, don’t be confused or intimidated by photographers who throw around the term ‘reciprocity’ – it’s not that hard of a concept: double the shutter speed, stop down one stop, half the shutter speed, open up one stop.
Naturally, changing the f-stop and shutter speed does change the image. Depth of field and the sense of motion are definitely affected by these changes. That sounds like another discussion…