First of all what are the bars of the saddle and what is the difference between the different kinds? Are certain ones more suitable for different builds of horses?
I like this website because it describes everything very well. Western Saddle Fitting and Different Tree Sizes
Basically, the bars of the tree are what rest on the horse's back (on either side of the spine).
Bars that are set wider apart (full quarter horse bars) will fit a wider and stockier type of horse, like a big stocky QH. Bars that are narrow-set (semi quarter horse) will fit narrow build horses.
All trees will have a slightly different shape and flare. One company's full QH bars will not be the same as another company's full QH bars. If the bars do not lay nice and flat on your horse's back, that will cause pressure points and pain.
Saddle fit is literally trial and error until you find one that fits your horse's back.
Also what is the rigging and how will I be able to tell which one is correct for my horse?
This website has a great description (with pictures) of what rigging is. It's all about where the cinch hangs down on the saddle, in relation to the saddle. Your Complete Guide to Saddle Rigging
All horses have slightly different anatomy to where their withers are and where the saddle "fits". So you may need to rigging farther forward or farther backward to get your cinch in the correct place.
Some saddles come with adjustable rigging, and gives you more freedom to adjust as you need for your horse.
What kind of saddle works best for a high withered horse? My horse doesn't have excessively high withers, but the way she is built, the saddle has to sit farther back to avoid hurting her shoulder. I've tried two different western saddles and both hit her withers. Is it true that barrel saddles tend to have more wither clearance?
In general, you are going to want narrower bars in order to "vault" over your horse's withers.
If the saddles you have tried so far hit the withers, you need a narrower gullet. (although a narrower gullet doesn't always mean a narrower tree)
I have never heard that about barrel saddles. A saddle is a saddle. It depends what tree is in it.
Speaking of shoulders I heard that the western saddle should sit partially on the shoulder, unlike the english saddle. Is that true?
Depends on the tree.
Most western saddles should NOT sit on the shoulder. It should sit behind the shoulder.
But a few brands, such as Caldwell with a rolled-edge tree, are designed for the saddle to sit partially on the shoulder.