There are a couple candies like it, they have a sort of chewy taffy like texture but are mostly sugar and corn syrup. How exactly do they cook or process it to create that?
Using something known as confectioner’s glaze.
It’s a simple form of soft candy cooked to a lower temperature than hard candies. These are sometimes called jellies but that term is usually applied to pectin based candy, which Skittles may be, I don’t know. Careful control of the proportions, temperatures, agitation produce a specific texture.
Heh 
Not used so much anymore. People don’t like to find out their candy is coated with bug juice.
The three main ingredients are sugar, corn syrup, and hydrogenated palm kernal oil.
No bug juice, but carnauba wax instead.
Major ingredients by volume, but I’ll bet you’re going to find stabilizers, artificial flavoring, and artifical coloring in there as well.
Yup, but everything aside from those three makes up less than 2% of the total volume.
That’s to get the shiny coating, not the chewy centre. It’s the bugs that have the soft centre
Back in the Olde AOL days, on the SDMB there was a thread about a theoretical war between m&ms and skittles. It was pretty close until the warring sides made alliances. Jelly beans, Gummy Bears and licorice allied with the skittles and things looked bad for m&m’s until Hanukkah gelt, Hershey Kisses, Snickers, Malted Milk Balls, 3 Musketeers and others joined the battle on the side of Goodness. Caramels sort of sided with whichever side seemed to be winning a particular battle. Ultimately Truth, Justice, Freedom, and Reasonably Priced Sex prevailed and the Skittle foe was defeated.
Eventually, The United Munchies Security Council sent in a peace keeping force of Circus Peanuts, Marshmallows, and Peeps. Things have been pretty quiet since then.
Manny People ask, so now I wonder, what happened to the raisinettes that were “resettled” by the Skittles.
Standard Skittles variants don’t use confectioner’s glaze to get that shiny coating, it’s carnauba wax as grude alluded to. Some of the variant/limited edition ones (IIRC the ice cream and chocolate ones) were non-vegetarian, but I can’t recall if that was due to confectioner’s glaze or gelatin.
Unfortunately for me, Junior Mints still use confectioner’s glaze for their sheen. Used to be one of my favorite movie candies.
Doesn’t matter, those stabilizers can be minute amounts and have a big effect on the texture.
Jesus Christ guys, how could you forget rainbows?
That’s the artificial flavoring. You really think they use real rainbows? 
Who the hell would want to eat candy made from unicorn farts?
Oh man, now I’m craving Skittles.
And you’re probably right, I guess I was thinking that less than 2% wouldn’t that much effect on the texture.
I know like you said the real gummy candies are either mostly sugar and gelatin, or pectin.
Well, yeauhhh!