Hi, Icelandic person here!
The Yule Lads' (that's a direct translation, the meaning is closer to "Christmas men" in Icelandic (and we also call the american Santa by that name)) mother, Grýla, is a troll/giant woman who used to eat naughty children just before Christmas (They say she's on a new diet these days, or dead, we're not sure). The Yule Cat, who is her and her family's cat, just ate kids who didn't receive a new item of clothing for Christmas, and naughty children didn't so in the end it really comes down to the same thing.
Grýla also has hundreds of other children, some of whom were Yule Lads in different parts of the country (My favorite "rejected Yule Lad" is Faldafeykir, who would create gusts of wind to blow up the hems of women's skirts. It is unclear whether he did this so they'd get cold in the freezing winter or if he was just a pervert). The ones we now recognize were handpicked by author and poet Jóhannes úr Kötlum, who published a lengthy poem about them in 1932.
Nowadays instead of causing trouble they travel to human settlements in the last 13 nights before Christmas, one each night, and leave gifts for children - the kids must place their shoe in a window, and that is where the gifts appear.
For those interested, the order is: Stekkjastaur, Giljagaur, Stúfur, Þvörusleikir, Pottaskefill, Askasleikir, Hurðaskellir, Skyrgámur, Bjúgnakrækir, Gluggagægir, Gáttaþefur, Ketkrókur, Kertasníkir.
Fun fact: all the Yule Lads were searching for food in some way, and this was what made them such nuisances when food was short (One would drink the milk from the sheep, one would steal all the burnt food remains that had stuck to the pans, and yes, Candlestealer would eat the candles). The only exception is Hurðaskellir (Doorslammer) who was just being an asshole.