What is the significance in an alpha-helix being right-handed or left-handed? Why is that often when alpha-helices are discussed, it is also mentioned their direction - right-handed (clockwise) or left-handed (anti-clockwise)? I have heard that left-handed alpha-helices are usually less stable, but I'm sure there is more to it than that?

Some experimental and modelling observations suggests, folding energy for right handed in more favorable. You can find more detailed answer to this question here.