The energy let-through graphs (as distinct from the more common time/current graphs) usually just show “energy” (I²t) a protective device allows to pass, given any particular fault current.

Personally, I never found the units I²t nor the term “energy” particularly obvious in this context - some here have suggested that Joules/Ohm might be a less opaque unit to choose - but suffice to say it's a useful quantity when working out how much hotter a conductor will become when exposed to a fault current before a particular protective device breaks the circuit. The other side of the equation being the “withstand” of the conductor - k²S² (which relates various thermal properties of the cable, and its cross-sectional area to the amount of “energy” it can safely absorb). You'll find I²t ≤ k²S² in various forms all over the place - e.g. S ≤ SQRT(I²t)/k.

Applying the same theory to melting of fuse elements gives you a basis for discrimination between fuses (but some care is needed because a fuse won't immediately break the circuit when its element melts - there can be a period of arcing where the element is broken but the current continues to flow - so there's both pre-arcing I²t and overall I²t values to consider).

Circuit breakers follow the same general principle (including an arcing period between de-latching and fully breaking the circuit) but the detail can be a little more involved as the energy let-through is far from constant, in fact often increases with increasing fault current.

As to all the whys and wherefores - some manufacturers have produced some very good guides - especially MEM/Eaton - they did a “The Guide to Circuit Protection & Control” which covered a lot of ground with relative clarity - unfortunately I've not been able to find a link to a publicly accessible copy. You might be luckier with Google that I was though.

   - Andy.

Please watch this clip in its entirety. The teacher is excellent.