Santa is a portly, white-haired and bearded old man that typically dons a red suit and pants with white trim, accompanied by a leather belt, red toque with a white-tufted tip and black boots and gloves, adapted for cold weather.
In accordance to the nature of Christmas, Santa is known as an effervescent, generous, and jolly person that gives bountiful gifts to the "nice" humans each year on Christmas Eve, going across the world on a magical sleigh. He's often seen as a positive person and known to be quite patient.
Role in Invader Zim
Santa is an extremely respected and blindly worshipped figure in the Invader Zim equivalent of human society, and most humans would gladly obey what Santa "wishes", with no regard if it would harm them or not, though Dib, his sister, and father are the sole exceptions. Zim once used this to his advantage by disguising himself as Santa and pretending to have "returned for all the good little boys and girls", allowing him to control the near entirety of the human race with ease. However, it seems that Santa is not an entirely mythical character; when Membrane was still a child, he received an endless supply of socks from the fat, bearded man; this caused him to turn his back on Santa and devote part of his time to eliminating him.
In Issue 16 of the comic series, during one of Zim's attempts to scare Dib, he has Willy dress up as Santa. Apparently, Zim thinks that "Christmas Man" is a figure of fear when he's present anytime other than Christmas. However, Dib's lack of reaction proves this to be a mistake on Zim's part. This would imply that Zim forgot about the time he dressed up as Santa and mixed him up with a figure of dangerous authority rather than the gentle, kind man he is known for.
In Quarterly 3, the third story features the origins of Professor Membrane's hatred of Santa, and how he dedicated years to figuring out how to hunt and fight him. Interestingly, where the Christmas episode of the show depicted Santa as being believed in and worshipped by all of humanity, in this issue Membrane is mocked for believing in Santa, and even occasionally berates himself for the "unscientific" belief.