Public Clubs are available to join by selecting the Clubs tab in your mobile device and tapping on "Find me a Club." With this option you will be assigned automatically to a Club.
To join a private Club, you will need a Club code from a friend or fellow language learner, who is learning the same language on Duolingo as you.
Once you have a Club code, open the Duolingo app on your mobile device, go to the Clubs tab, and tap "Club code." You will then be prompted to enter the Club code.