Apps & Invites will display any clan who has sent you an invitation, you can choose the appropriate icon to accept or deny the invitation.

Below that you will be able to submit an application to any clan that allows them, just enter the name of the owner and click Submit and your application will be submitted, if the name you enter doesn't have a clan or their clan can't accept any applications, you will be notified with an error message. When your application is accepted by the clan owner or one of their mods, the clan will immediately appear in your Current Clans list and you will appear on their Clan Members list. You can withdraw the application here also if you want to reclaim your Clan Slots.