Monkey - Learning. Many newbies start here.
Tiger - Bold. Defeats monkeys by being more aggressive.
Leopard - Tactical. Defeats tigers by being more organized, controlling his units better etc.
Crane - Patient. Defeats leopard with strategy such as traps, better econ, etc.
Dragon - Expert/God. Knows everything about the game, ready for any map. Defeats the crane by being ready for anything, bringing a mix of units (like STs with his CAs for example, or a cav/camel/CA combo).
Snake - Paranoid. Knows he's probably going to get killed (usually does), but won't go down easy. Reacts to his opponent's game in irrational ways, attacking his plan instead of his army. Weakens the attackers enough for his allies to win, even if they would normally be weaker. This upsets dragons greatly, hurting their self-discipline even more.
Monkey - Learning. Usually gets a bit rusty from being less competitive and experimenting with weak civs, weak units, etc. Just playing for fun, doesn't expect anything, with no specific plan for snake to mess up.
Loopy loop. There is no best playing style, and if you play long enough you'll probably go through the full cycle several times.
In other words, I agree with popov. There's always more to learn, you can always improve.