What types of mountain bike handle bar grips are available and how to they affect comfort and handling?
Which grips you get for a mountain bike is mostly a matter of comfort and personal taste. 

In terms of how they attach to the bike, there are grips that glue on and grips that screw on. They're exactly what they sound like. Locking grips (that tighten onto the bars with an allen key or a screwdriver) will avoid the grips twisting over time. 

You can also get "ergonomic" locking grips. I had a pair of these and found them quite comfortable, and again, this is a matter of personal preference. (I also found that they reduced fine control just a little bit.) Some even come with smaller barends built-in.