To leave a Clan, first navigate to the Clans section of the game by tapping the Clans icon on the flock screen. You will now see the information for the Clan you are a member of. Tap on the "Clan Info" button, and you will then see a red "Leave Clan" button. Tap that and you will be given a confirmation window asking if you really want to leave the Clan. To confirm, simply tap the red "Leave Clan" button again!
Please note that clan leaders can only leave their own clan if they first promote a co-leader to leader.