Tantalus, in Greek myth, was a king of Lydia and a son of Zeus — one of the few mortals invited to dine with the gods. He repaid them by stealing nectar and ambrosia from their table, and worse: in some versions of the story, he killed his own son Pelops and served him at a banquet to test the gods' omniscience. They were not amused.
The Punishment
His punishment, described in the Odyssey, was endless and exquisite. He was placed in a pool of clear water, with branches of fruit hanging just above his head. Whenever he stooped to drink, the water receded. Whenever he reached for fruit, the wind lifted the branches out of his grasp. Hunger and thirst, eternally renewed, with their satisfaction always just barely visible.
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A Verb in the 1590s
The English verb "tantalize" appears in print in the late 16th century, formed from "Tantalus" by adding the verb suffix -ize. From the start it meant exactly what the myth described: to torment a person by holding something desirable just out of reach. Shakespeare uses the word; so do most English writers from Marlowe onwards.
Tantalum, the Element
The metal tantalum is also named after Tantalus — for a charmingly literal reason. The 19th-century Swedish chemist Anders Gustaf Ekeberg, who isolated the element, found it remarkably difficult to dissolve in acid: it would not "drink." Surrounded by acids and yet unable to absorb them, it reminded him of Tantalus standing in water he could not drink. The element kept the name.
The two senses of tantalize — the verb and the metal — both name the same predicament: surrounded by what you want and unable to take it.