Clans can be set to "Invite only" by the clan leader or co-leader, this helps better control who is able to join. You can ask to be admitted to the clan (if you have enough trophies) by sending a request.  After sending your request, it will appear in the clan chat. Only the leader, co-leaders, or elders have the power to accept or reject your application.

As long as your request is showing in the clan chat, you will not be able to send a new request to that, or any other clan. You must wait until your request disappears or is rejected, or until someone else invites you to a clan directly.