Effective aperture is relevant solely to macro photography, and as a rule of thumb, you need not care about effective aperture until your magnification reaches roughly 1:2.
Effective aperture results from that most macro lenses work by moving the objective (the entire constellation of lens elements) away from the film/sensor. This not only allows the lens to focus closed, but it also narrows the field of view of what reaches your film/sensor and ‘effectively’ changes the focal length. As the physical opening in the diaphragm however remains the same, this leads to that the aperture value (defined as focal length/diameter of physical opening) decreases. This relationship follows roughly the formula of:
f-stop x (1 + Magnification) = Effective f-stop
In other words, when you set your 1:1 macro to f/5.6 and focus down to MFD, your effective aperture becomes … wait … f/5.6 x (1+1) = f/11.
Effective aperture is important, because it’s not just something theoretical – the aperture value (focal length/opening diameter) really does change. It hence effects not only the amount of light reaching your film/sensor (and on a modern camera you might have noticed that the metering values your body gives change when you focus closer), but also introduces the bogeyman of everyone hoping to take pin-sharp pictures: diffraction. Hence, it is rarely advisable to go beyond f/8 in 1:1 macro (because that effectively is f/16 and is already diffraction-limited).
Now, you might now think: ‘Hold on, doesn’t this mean that whenever focus breathing occurs, the effective aperture also changes?’
And you would be entirely correct. Any f/2 50 mm lens, when focused to MFD and now being effectively a 60 mm lens would also effectively be an f/2,4 lens, but convention has it that effective aperture calculation is disregarded at anything below 1:2 magnification.
Most modern camera and lens combinations are able to take effective aperture into account. In practical terms, this necessitates only that the lens is able to communicate its actual focusing distance to the body, which then calculates effective aperture and adjusts the exposure accordingly.
Importantly, in the era of manual lenses, most camera bodies had no way of knowing the focusing distance, and could not think on the photographer’s behalf. In practical terms, if the photographer was using the camera’s TTL metering (if that was available), the results would be quite okay, but if the photographer was relying on a hand-held meter, calculating the effective aperture was a ‘must’ to be able to take well-exposed pictures.