Genie Magic can only be used by genies, who have the power to grant three wishes to any human that picks up their residing lamp or bottle. There are, however, some wish restrictions. One cannot wish for another person's demise by death, bring back the dead, make someone fall in love, change the past or use one wish to ask for many more wishes. ("Fruit of the Poisonous Tree", "Trust Me")
While the genie grants the wishes they happen once the person says "wish" not actually having the genie to do anything themselves. Also, while the wish is granted the genie has no control over what happens, therefore if the wish is not specific it can have dire consequences. Once the wishes are granted the genie goes back into the bottle for the next person to use them.
Genies are injured when they come into contact with silver, which burns their skin. When Cyrus was imprisoned by Jafar, he was locked inside a silver cage to keep him from escaping. ("Trust Me")
The tradition of breaking a wishbone so the one who gets the larger half gets a wish, is a story created long ago by genies to make people believe that they do not have to seek out a bottle to get what they desire. ("The Serpent")
Similar to genie magic, the Wishing Star has the ability to grant a wish to its user. However, they are dissimilar in that the Wishing Star needs the user to be pure of heart to obtain a wish while genie magic only requires the user to be the owner of a genie's lamp or bottle. ("Fall")
↑The exact spelling varies, depending on the translation. This article uses the version found in the first edition of the "Aladdin" story, a story which was not found in the original Arabic manuscripts of One Thousand and One Nights, but was set down on paper for the first time by Antoine Galland (the French orientalist who was the first European translator of One Thousand and One Nights) in 1712; Galland heard the tale from a Syrian storyteller. (Source) The first edition, which can be read here (scroll down to "HISTOIRE D'ALADDIN, OU LA LAMPE MERVEILLEUSE"), uses the word "génie", which is French for "genie".
Note that many publications cite Richard Burton's English translation of One Thousand and One Nights (source); his translation of "Aladdin" (published in the third volume of The Supplemental Nights to the Thousand Nights and a Night with Notes Anthropological and Explanatory from 1887), which can be read online here, uses "Jinn" and "Jinni" instead of "genie".