scanf("%s", &word[0]);
printf("%s", word[0]);
In C, a string (addressed by %s in above statements) is defined as an array. Your declaration of "word":
char word[1];
declares an array, so that array can be a string (I wrote can, because it's a very short string...). If you use the identifier alone, "word", the compiler uses a pointer to the first element of the array. Hence:
scanf("%s", word);
printf("%s", word);
are both correct. But if you use an index into the array, like "word[0]", then you are no more using the array per se, but instead an element of it. This notation does not generate a pointer, but your above statements do need a pointer. So you can use an "&" to get a pointer.
Conclusion: if you have a string, use its simple name without indexing inside the array. Using the ampersand makes it clear that we pass a reference, but "&word[0]" is ugly and, moreover, everybody should know that scanf() writes into its arguments.