Well, I have three girls, so know a little about pig tails. Pony tails are what you are describing. Longer, thicker hair pulled together into one "tail". Pig tails are usually found on little girls whose hair has not grown long yet, so their "tails" may only be 3 or 4 inches long, where a pony tail may be 12 inches or 24 inches long.

Pigtails are when you tie your hair with two elastics, one on each side of your head. It doesn't matter how thick or thin or how long or short your hair is, it's still pigtails. Ponytails are when you tie your hair with one elastic on the back of your head. (And a "side pony" is when you tie your hair with one elastic on one side of your head.) If you put your hair in two braids, that's also commonly called pigtails, but if you put your hair in one braid, it wouldn't be called a ponytail. Here's a picture I found on Google with an example of pigtails: http://www.prettydesigns.com/wp-content/uploads/2013/12/Nicole-Kidman-Long-Hairstyle-Playful-Pigtails.jpg