In a typical lottery, a person pays to participate. I would go to a local convenience store, pay some money, win, and gain great happiness. In the story "The Lottery" by Shirley Jackson, most of that is completely flipped around. A person automatically gets to participate in the lottery. It's free! If you win. . . then you have to pay the price . . . with your life . . . and great happiness doesn't spill forth from the winner's mouth. The "winner" gets to be stoned to death by the remaining "losers," so I imagine instead of cries of joy, there are screams of agony and pain. A person doesn't win money; they win death.
Another key difference is that in a typically lottery, people get to choose whether or not to participate in the lottery. In Jackson's story, every single member of the community must have their name in lottery box. The only exception to that rule is Mr. Summers.