Think of a number. Any number at all... between 10 and one billion.
Now, since that number has at least two digits, add up all of the digits, and subtract that sum from your original number.
Next, add up all of the digits of the number you just got after that subtraction, to get another new number. And then do it again with THAT number (if you have only a one-digit number, you would just get that same number again).
Finally, subtract 1 from that last number, and find the letter of the alphabet that corresponds to the result (1=A, 2=B, 3=C, etc.) Now think of one of the 50 United States that starts with that letter.
Aloha! You've just arrived in Hawaii, didn't you?
How does this work? The main "trick" to this is knowing that when a number is divisible by 9, the sum of its digits are also divisible by 9. Therefore, once you have a number that is divisible by 9, you can keep summing up the digits, and eventually end up with 9 as the final result. For a number less than one billion, the sum of the digits can never exceed 81 (for 999,999,999)... therefore you'll only have to do the digit-summing twice (at most) before ending up with 9 as the result. Once you end up with 9, then subtract 1 to get 8, this leads you to the letter H, and Hawaii is the only choice.
So, the only remaining question is how to force the person to a number that is divisible by 9 in the first place. That's where the first part comes in: subtracting the sum of the digits from the original number. This works because our numbering system is based on multiples of ten, so subtracting one of each digit leaves you with multiples of nine! Here's an example:
The number 3198 is the same as the total value of 3*1000 + 1*100 + 9*10 + 8. Subtracting the sum of the digits (3 + 1 + 9 + 8) means that we can remove one 3, one 1, one 9, and one 8 from that total. That leaves us with 3*999 + 1*99 + 9*9. Notice how each of these terms is now divisible by 9; therefore, the total will be as well!
That's the whole trick. Force the person's randomly-chosen number to something that's divisible by 9, then use the digit-summing to reduce it down to the number 9. After that, the rest of the trick is just for flair. Instead of subtracting one and asking for a state, you can subtract 4 and ask for a zoo animal (Elephant, though some may think of emu)... or subtract 5 and think of a country (Denmark, though some may think of Dominican Republic). There are all sorts of variations on this trick, so make it your own and impress your friends!