In C, a string, as you know, is a character pointer (char *). If you want to swap two strings, you're swapping two char pointers, i.e. just two addresses. In order to do any swap in a function, you need to give it the addresses of the two things you're swapping. So in the case of swapping two pointers, you need a pointer to a pointer. Much like to swap an int, you just need a pointer to an int.
The reason your last code snippet doesn't work is because you're expecting it to swap two char pointers -- it's actually written to swap two characters!
Edit: In your example above, you're trying to swap two int pointers incorrectly, as R. Martinho Fernandes points out. That will swap the two ints, if you had:
int a, b;
intSwap(&a, &b);