The hand seals in the Naruto series are based off of the Oriental Zodiac system. As you may have heard when someone talks about what year they are born in, they may say year of the Dragon, which means they would have been born in 1928, 1940, 1952, 1964, 1976 or 1988. Below we have organized the hand seals in order of when they appear in the Chinese zodiac, from one to twelve.
Nezumi (Rat) 
O-ushi(Ox) 
Tora (Tiger) 
Usagi (Hare) 
Ryu (Dragon) 
Hebi (Serpent) 
Uma (Horse) 
O-hitsuji (Ram) 
Saru (Monkey) 
Tori (Bird) 
Inu (Dog) 
Ousu-buta (Boar) 