There are literally thousands of resources talking about clicker training and the principles behind it, so I'll just cover some basics you need to look into.
Firstly, clicker training was originally designed to use with whistles when teaching dolphins and whales tricks. The method goes back further and you'll run across Pavlov's dogs. We actually learned about him in school before I learned about clicker training. I think it was Karen Pryor who started adapting it to dolphins and whales.
It's based on a phenomenon know as operant conditioning. Basically, this means that you teach the animal (and it works on people as well) that a sound, action, stance, etc... equals something else. With Pavlov, he would ring a bell and the dogs would start drooling, because he always fed them after the bell was rung. So with no food in site, their mind made the connection. If you want a real life example, think about how a cat will come running if they hear anything like a can of wet food opening. It might not even be food. It could be something inedible that just sounds like the food can opening. Another example would be my dogs getting hyper excited anytime I put my tailgate down on my truck, because 90% of the time I load them up and take them to the river. People do this as well. Do you ever have someone call and you don't want to answer it, because you know they usually call to ask you a favor? You don't really know, but your conditioning is that they'll probably want you to do something for them.
So that's what you're doing with clicker training. You're associating an action with a reward. The click, which can also be a whistle or just your voice, marks the behavior and is hence called a 'marker'. It is known as a 'bridge', meaning that in the animals mind, it bridges the gap and connects the sound of the clicker to the reward and also to the action that caused the click.
You may be asking why you can't just say 'good boy'. In reality you can. The benefit of the clicker is that it's consistently the same sound, length, and duration. It's also short and sharp. This allows your pet to know the exact thing he was doing when he was clicked and rewarded. In traditional training, you'd get a behavior, say "good boy", and walk over and treat them. Even in the time it takes to say "good boy" your dog could have done a multitude of behaviors. You may be trying to get a sit, with a clicker, you mark the instant his butt touches the floor and he'll quickly associate that butt touching floor equals click equals treat. However, with the traditional method, he could have touched his butt to the floor, started to sit up, looked left, opened his mouth in excitement, etc... So it takes much longer this way, though you can probably achieve the same results. The down side of a clicker is you don't always have one on you, though I consider it a minimal downside as training sessions are usually planned.
As you picked up, some people don't always use treats. This is for a variety of reasons. Some dogs actually prefer affection or toy play to treats and you'll see a lot of schuntzhound (probably mangled that word) and protection dogs that are more motivated by toys than treats. Most of the time though, it's probably because they've started phasing them out. You won't always need the treats of the clicker. They're for establishing the behavior. Once it's ingrained, you can start asking for multiple repetitions or chained behaviors before clicking. This is how you do complicated tricks. You train small behaviors and link them into one big behavior.
Over time, you'll be able to phase out the treats almost entirely and you'll be able to phase out the clicker much sooner. You will need to treat them periodically. They eventually come to see that their effort isn't worth the lack of reward if you don't keep them hoping they'll get one sometime. It eventually becomes ingrained as the thing to do. It's potty training a kid. The parents have to tell them over and over that they need to try to use the big boy potty. They don't understand the words, but eventually get the intent and I imagine that no one reading this still thinks it's a good idea to potty in their pants. But hey, I could be wrong. :D
So that's the gist of it. I'll go into a few pointers now. Firstly, sessions should be short. Your animal has almost no attention span. It'll grow after a long time of training, but if you go too long at first, they get frustrated and shut down. Multiple 5min training sessions throughout the day are easier on you and the dog than one 1hr session. Also, don't try for too much at once. If you get some good results, quite there. If they get confused, back up to something easy, quite for the day, and think about how you can help them get there. 1% progress a day is 100% improvement in 100 days.
Ways of getting behaviors you should look into are luring, shaping, and random capture. Random capture is sometimes the only way to get what you want. Also, chaining behaviors is important to advanced trick.
Lastly, you're talking about training a cat. While it's definitely possible, prepare to be frustrated. A dog wants to please you, a cat doesn't give a crap. It's strictly reward for action. Kind of like people won't work if they aren't paid. So keep the sessions short and don't worry if they just quite or walk away, they'll get their eventually. The easiest way to start with a cat is a plastic spoon taped to a rod. I like the telescoping rods for storage. Put a dab of wet food on the spoon and the cat will naturally sniff it. When they touch it with their nose, click and treat. When they understand the bridge, try getting them to touch it without food on it, click and then treat. It's important to get the behaviors consistent before adding a verbal cue and you shouldn't use a verbal cue more than once. If you have to, then the behavior isn't good enough for a verbal cue. The behavior I'm describing you train a cat with is the basis of a 'Touch' command and will eventually allow all sorts of behaviors like, 'come', jump up here, fetch, etc...
I hope this info helps you. Good luck.
