After Thor's banishment in the first movie we find out how Odin took pity on a frost giant baby who turns out to be Loki. It is also found out he is the son of Laufey who is the ruler of frost giants. Assuming he is a pure breed Frost Giant why isn't his son's skin blue. We see it briefly blue when Odin lifts Loki as a infant and then when Loki lifts the Casket of Ancient Winters. So why doesn't Loki share the color of his race and that of Asgard.
Odin changed his appearance. You can see in the movie that as an infant Loki's skin was in fact blue but when Odin picked him up, he rubbed his face and changed his appearance. It's Odin's active power that keeps Loki's appearance as we see him. Odin was making plans for Loki's future as a bridge between Jötunheimr and Asgard even then.