First the clicker:The purpose of the clicker is to provide a "bridge" between the behavior you're trying to reinforce and the actual reward. This enables you to "mark" specific behaviors by clicking simultaneously with them when it would be impossible to give them a reward for the behavior right then.
A clicker works very well as a bridge because it is a distinct and consistent sound that creates a strong, clear association between behavior -> marker (the clicker sound) ->reward. A sound works better than any other type of reinforcement because it will pretty much always be noticed and recognized.
However, any sound that is distinct and consistent will work for "clicker training". For instance, I know some people use the caps from Snapple bottles (they click when you push them in) and PunksTank uses a smooching sound and doesn't even have to carry a clicker device! The key to a good "marker" is making sure that it's always the same and always associated with your reward. For instance, if you make a smooch noise for a cue, then a smooch noise will not be an effective marker since it's not clear what you're indicating when you make the noise. Moreover "good boy" or "good girl" is usually a poor marker choice because you're likely to make the same words or even just the sounds in other contexts and confuse the horse, and even our best efforts to say this the same way every time will likely fail since things like emotion will affect how we say it. Personally, I don't trust myself to be consistent enough with any verbal cue, and so I have my clicker permanently attached to my wrist with a high-quality elastic wristband and it's just one of the pieces of tack I grab when I intend to work with my horse. If I can grab a halter and lead rope, I can grab my clicker :)
Now, the treats:
Once you understand what a reinforcer really is, you can decide what you'd like to use as your reinforcer. As long as it motivates the horse to work, it is a reinforcer! Preferably, you want a reinforcer that the horse will work for over a period of time as well. Does your horse work for a scratch behind his ears? If you'd like, you can use that instead of treats! However, treats are often the most convenient reinforcer for a number of reasons. First, most horses are food-motivated simply because it's a basic need, so we can exploit it. Not all food will work for all horses - for instance, one of my horses only likes a few bites of grain and then loses interest. Grain would not be a good reinforcer for him, while it probably would be for most horses. I like using "cookies" because I believe they're healthier and I don't have to worry about him getting too much. Plus, I can change flavors to keep him interested. I try to find the smallest ones I can so that I can give a small reward without feeding too much each time.