Question on fuses specifications
What does In (whatever the right spelling may be) mean in this context?


It is \$I_n\$ and it indicates that the measurements were taken at 10x the nominal current. The best example might be the 30 A fuse. A rating of \$1000 A^2 s\$ means that at 300 A it should blow around 0.011 s. That lines up roughly with the graph.


  What does the Time-Current characteristic curve tell me? If I look at the 10A curve [...] then 20A maps to 5 sec. Does that mean that if a fixed current of 20A runs through it, after 5 sec it will blow?


Yes.


  What does the Amp rating really relate to?


The amp rating on the face of the fuse is the nominal current. A 10 A fuse can operate indefinitely at 10 A. However, as you've seen, this 10 A fuse can also operate at a steady 15 A for ~1.4 hours. All fuses are designed to open during a short circuit but the time-current characteristic will give you an idea of the rest of its operation.

Try looking at the time-current characteristics of "fast-blow" and "slow-blow" fuses. You'll find fuses that can operate at 1 A indefinitely, but will blow at 2 A within 0.1 s. It all depends on your application.