Basic Definition

Twill is a cotton cloth woven in such a way to produce weave with a thick rib (line) that runs diagonally from one corner of the fabric to another. Denim is a type of twill-weave fabric.

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Right Hand Twill

According to “the anatomy of a jean,” by Oki-ni, right-hand twill is a weave which produces a diagonal line which rises from left to right. This was standard practice in weaving. Typically, the single yarn warps were woven right hand and double yarn warps were woven left hand.
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Left Hand Twill

The left hand twill or the ‘S Twill’, this is a weave in which the grain lines run from the top left-hand corner of the fabric towards the bottom right. Usually in piece dyed fabrics, left hand twill fabrics are woven from single plied yarns in the warp. Left-hand twills will often have a softer hand feel to them after washing than right hand twills.
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BrokenTwill

According to an article from Cheng’s measuring method titled All About Denim, instead of the twill running to the right or left, broken twill jeans contain no distinct direction of weave. The weave is instead alternated right and left - the end effect resembles a random zig-zag. Wrangler made the first broken twill jeans in 1964 after it was invented by John Neil Walker. Broken Twill was designed to combat the twisting effect that was a characteristic regular twill. By going on both directions, the tension in the yarns is balanced in Broken Twill. The broken twill is also traditionally considered the cowboy-preferred denim.
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See Also

denim
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References

Anatomy of a jean, http://www.oki-ni.com/page/anatomyofajean

Cheng’s measuring method, all about denim. http://www.chengwebsite.com/cmm_denim.html
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