Clicker training is a positive reinforcement training method based on operant conditioning. The emphasis in clicker training is on shaping behaviors you want through positive reinforcement rather than eliminating behaviors you don't want through punishment.
Clicker training doesn't really depend on the clicker or food. It depends on the reinforcers or rewards which are anything the dog likes. The clicker and food are training tools which you replace with a word and a pat once the dog understands the behavior you want, how to do it and when to do it.
The click lets the dog know what it just did was right. The click marks the behavior and ends the behavior. So, if you click the puppy for a good sit and it stands for the treat, that's okay. It doesn't matter what the puppy is doing when you give the treat. Your puppy will remember what it was doing when you clicked. It is important to remember the click always comes first, followed by the treat. Never click and not treat!
You can begin clicker training, however, at any age in a dog's life. It's fun, gentle and effective.