The first step in training your cat is finding a way to communicate to them what behaviors you like. One way of doing this is to associate a signal (a neutral stimulus) with a natural response. Remember learning about Pavlov’s dogs in psychology? Ivan Pavlov found that an initially neutral stimulus (metronome- although it is widely believed he used a bell!) can be paired with food to elicit a natural, reflexive response (salivation). After multiple pairings with food, the metronome click by itself became enough to elicit the salivation response. Upon hearing the metronome the dogs “anticipated” the food would soon be delivered and began to salivate in response to the metronome alone. This is the basic concept of classical conditioning and an important aspect of animal training.
In animal training people often use clickers to produce a neutral “click” sound. There is nothing special about the “click” until it is given meaning (conditioned). In reality you could condition almost any neutral stimulus (I’ve heard people use a harmonica, bell, snap of their finger and even light for blind animals) but clickers often produce a consistent, distinct sound which makes training with it simple. The idea of the “click” is to mark the behavior you like and want the cat do more frequently. So, first the “click” must be associated with something good (as the metronome was with the food). The click essentially tells the cat, “I like what you did, you’ll get rewarded for it soon”. This is called a bridging stimulus because it marks the exact moment of the behavior you like- bridging the time between that behavior and when the animal is rewarded.
Decide what your reward (or reinforcer) will be! A reinforcer is something you give to your cat that will cause an increase in the desired behavior. This will be something your cat finds rewarding and wants to work for. You can use food, praise/attention, and play as a reinforcer. For food I suggest trying cooked chicken, tuna, or summer sausage. See this post for more information on picking the correct reinforcer for your cat. This is very much on an individual basis- your cat will decide what is reinforcing for them and each cat will be different. It will be hard to ask your cat to work for an item they do not find rewarding, so make sure you choose the correct reinforcer!
Pair the click (or whatever stimulus you have chosen) with your reward, or reinforcer. If you don’t do this the cat will not understand what the click means. To do this, simply click and then give your cat the reinforcer. Do this multiple times, ~30x with small amounts of the reinforcer (if using food use small pieces, if play/petting do short bouts), until the cat associates the click with the idea that a reward will be delivered soon. You will start to notice the cat expect something when they hear the click. For example you may click and the cat will immediately look up at you, waiting expectantly for their treat. Once this happens you know your kitty understands the meaning of the click. Then it is time to start training!