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In the US, state-run lotteries typically offer a choice of three or four games. Most include a numbers game, such as the Powerball, and a smaller number of instant lottery games (also known as scratch-offs). In addition, some states also run keno or other video lottery terminal games.
There are two key elements in any lottery: a way to record identities and the amounts staked by bettors; and a mechanism for selecting and distributing prizes. The former, typically a numbered ticket, has the name and address of the bettors printed on it; the latter is a receipt entitling the bettor to a prize if the purchased numbers are drawn. Modern lotteries often have a central computer system for recording purchases and generating winners, but older ones used a combination of secret codes, the names of bettors and the dates and times of their purchases.
State lotteries are intentionally run as businesses, and the commissions that run them have every incentive to tell voters all the good the lottery does by raising money for state governments. This is all true, but it overlooks the fact that a significant portion of that money is spent on marketing and that, in relative terms, lotteries raise a small fraction of actual state revenue.