What do you call the elf or god coming out from the Aladdin-lamp in English?

A Genie.

"Jinn" (variant spelling "djinn", singular : "Jinni") or "genies" (singular : "genie") are supernatural creatures in Arab folklore.

More stories of the Jinn can be found in the "One Thousand and One Nights" story of the Fisherman and the Jinni; more than three different types of Jinn are described in the story of Maruf the Cobbler; a mighty Jinni helps young Aladdin in the story of Alladin and the Wonderful Lamp.

Name not normally translated in the case of the djinn. I believe if it were translated it would mean demon. Although it doesn't include the tale of Aladdin I have a copy of the 1001 Nights that was properly translated and the word in English is translated as demon.