There are 2 parts to Current Clans, the sidebar category and the settings page.

The sidebar provides easy access to change your currently selected clan and links to any applications and invitaions you have pending. If you are the owner of your selected clan or a mod with proper permissions it also shows links to Clan Admin and any applications and invitations for your clan. If you are not in a clan the sidebar just shows a tip on how to find a clan you might like.

On the settings page you can set any clan you are in to your selected clan by clicking the round button next to it then clicking the Set Clan button. This will make the clan related sidebars show the name of the clan as entered by the clan owner and each clan specific page (i.e. Leaderboard, Clan Skills, Clan Admin) will show the data for the clan that is selected. You can leave any clan listed here by selecting a clan in the list and clicking the Leave Clan button.

You will also find any clan that has removed you or clan that was deleted by the owner while you were a member listed here. The only option avaliable for these is for you to delete them and reclaim your Clan Slots. You will not be able to be re-invited or re-apply until you delete these clans from your list.