Wool is going to be impossible to cover in one post, because not all wool is the same. There are, in fact, eleventy bajillion and one (or something... I can't find a definitive number) different breeds of sheep. Not all of them are wool sheep -- many are what is called a hair sheep, which are raised exclusively for meat, and do not need to be sheared because they shed their coats in the spring. The wool sheep all produce different wool. Some is coarse and scratchy, some is buttery soft, and everything in between! There are, however, a few things that all wools have in common.
Wool felts. Soft wools are generally easier to felt than course wools, but under the right conditions, all wools do felt. The only exception is superwash wool, which has been treated specially to prevent felting. Even then, though, some superwash wools will felt if conditions are extreme enough. The recipe for felting: agitation, water, heat, soap. Basically what happens is the tiny scales on the fiber's surface open up, interlock, then close up again. Felting shrinks wool severely and makes it denser. Felting is permanent.
Wool can be dyed with acid dyes. This means no icky mordants for the home dyer -- you can dye wool by microwaving it in a bowl of Kool-Aid! This means it takes minimal equipment to set up and try, which is always good in my book.
Wool has memory. That basically means that whatever position the fibers dry in, they want to stay in, until they get wet again. This is good for things like ribbing (it will stretch and spring back) and also for lace (when it dries stretched out it will stay stretched out). Some wools have more memory than others.
Many yarns just say "wool" on the label. This means that the company has a standard blend of sheep breeds for consistency within the line, but it's impossible to know the exact properties of generic wool without either touching it or more descriptors. For example, Patons Soy Wool Stripes is ridiculously itchy. Patons Classic Wool, on the other hand, is fairly soft. Both are labeled simply as "wool."
As this is becomming quite a long post already, part 2 will come later, and will discuss the relative benefits of soft wool and coarse wool. Yes, you may sometimes want to use coarse wool!