There are languages that index arrays defined for N elements in [1...N] (matlab comes to mind) or even [0...N] (Blitz Basic).
There is a good reason for the typical choice of [0...N-1], but it goes somewhat deep. You need to remember that the end result of the code is meant to run on actual hardware. In this case you want to access memory, but memory itself doesn't know the concept of an "array". It knows what data it holds and what index it's at. The most important thing is that an array is actually some index to the memory, and more precisely it's the index of the first member of the array. To access this, there is the typical array[j] syntax, which pretty much means "give me the j:th element of array 'array' " which is the same as finding the bit of memory that's at array+j. Since array is an index in the global memory and j is the index inside that array, we can just sum them to find the location of j in the memory.
Now, it (hopefully) makes sense to choose that the first element of "array" is at memory address "array", not at "array+1", so the first element should be at array[0] instead of array[1], since it gets translated to an actual memory address. Why it ends at N-1 instead of N is because we still want to have N elements - and if we count from 0 and go forwards until we have N elements, we reach element N-1.
This is, of course, only a chosen convention but it implies other nice things, like for(int i = 0; i<N; i++) /*operate on array; */ instead of for(int i = 1; i<N+1; i++) /*operate on array*/; the length of the for loop can instantly be seen from the ending condition instead of having to remember to take one out - it's a simple thing, but would cause (I believe) even more headache, especially for beginners or less enthusiastic programmers.