A variation of the word "hey." Can be used as a greeting, usually with the syllable drawn out while simultaneously rising and falling in inflection, as illustrated below in Example 1.
Can also be used as an expression of enthusiastic agreement in a conversation, as demonstrated below in Example 2.
One theory suggests that this variation of "hey" was popularized by the character of Dwayne (portrayed by Haywood Nelson) in the sitcom "What's Happenin'," although Dwayne's usage was three "hays" with shortened syllables, with the inflection on the middle "hay," as shown in Example 3 below.
Person #1: "It sure is hot today!"
Person #2: "HaAAAAAay!"
A bastardization of the greeting "How are you doing?" Used by the stereotypical mobster in popular culture. Heard in countless movies, TV shows, commercials, etc. Most often used in real life by actual mobsters, Americans of Italian descent living in New York or New Jersey (and elsewhere), or retards who want to sound tough.
Tony Soprano: (to his crew as he walks into the Bada Bing!) "How you doin'?"