I think originally Elf on the Shelf was supposed to come to spy on children and report back to Father Christmas on whether or not they have been good. Personally, I find that a bit creepy, so our elf just comes for a bit of festive fun and mischief! He arrives on 1st December, when he hangs and fills the advent stocking garland and a letter from Father Christmas. Then throughout December he does different things each night - sometimes he just hides somewhere, or he might bring a little present (Christmas DVDs or books usually) or supplies for a Christmas craft (homemade tree decorations, paper chains, paper snowflakes). Sometimes he plays or makes friends with the kids' toys (riding on toy trains / cars, building with lego / duplo, raiding the musical instrument box and forming a band with a group of cuddly toys). Some of my favourite days are when he gets up to mischief (raiding the christmas treats, wrapping himself up like a present, putting pants on the Christmas tree). Then on Christmas eve morning he brings new pyjamas and ingredients for reindeer food, and at bedtime we put him out with the stockings and treats, and Father Christmas takes him home.
Pinterest is a really good source of ideas, have fun x