Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or "drawing", cards.
The descriptions below assume the reader is familiar with the general game play of poker, and with hand values (both high and low variations). They also make no assumptions about what betting structure is used. In home games, it is typical to use an ante, and betting always begins with the player to the dealer's left. In casino play, it is more common to use blinds; the first betting round thus begins with the player to the left of the big blind, and subsequent rounds begin with the player to the dealer's left, thus draw games are very positional.
Some sample deals below will assume that a game is being played by four players: Alice, who is dealing in the examples, Bob, who is sitting to her left, Carol to his left, and David to Carol's left.
Standard five-card draw
This is often the first poker variant learned by most players, and is very common in home games although it is now quite rare in casino and tournament play. Two to eight players can play.
Other draw games
Gardena jackpots ("Jacks to open" or simply "Jackpots") Played as above, with standard hand values, and with a single joker in the deck acting as a bug. It is always played with an ante and no blinds. On the first betting round, no player is allowed to open the betting unless their hand already contains a pair of jacks or a better hand. Other players who checked on the first round may subsequently call or raise if someone else opens. If no player opens, a new deal begins and everyone antes again into the same pot. The player who opened the betting keeps their discarded cards near them on the table so that they can prove, if necessary, that they had a sufficient opening hand. For example, a player with the and the has a pair of jacks and may open. They may wish to "break openers" in this case by discarding the jack of hearts in an attempt to make the club flush, so they keep the discarded jack to prove that they were entitled to open.
In addition to the opening requirement, no one is allowed to win a pot with less than three of a kind. In the event no one makes the winning requirement, the pot is moved on to the next hand. The strongest possible hand in this game is five aces which is achieved if the player obtains the four aces plus the joker (joker counts as an ace if there's neither a straight nor a flush to complete).
This can result in some enormous pots, and is where the term 'Jackpot' comes from.
The game is named after the city of Gardena, California, where this game was especially popular from the 1930s to 1970s (though it was always secondary to lowball). At that time, there were more public poker tables in that small city than in all the rest of the United States. Public poker rooms are still a big industry there, though Las Vegas, Atlantic City, and other locations now have many more poker rooms than they did at that time. Because "Jacks to open" was the primary form of high-hand draw poker played there, traditional draw poker was often described by the retronym "Guts to open".
In home games, it is common that when a deal is "passed out" (that is, when no one opens), the players re-ante, and the qualifier to open is raised to a pair of queens. If that deal is passed out, the qualifier is raised to kings, and finally to aces. This is called "progressive" jackpots.
California lowball
This was the primary poker game played in California during the heyday of Gardena in the 1970s. It is still played today, though its popularity has somewhat lessened since the introduction of stud poker and community card poker to the state.
Played as above, using ace-to-five low hand values, with a single joker in the deck. Always played with blinds rather than antes, so players may not check on the first betting round (but may on the second round). A player with a 7-high hand or better who checks after the draw forfeits their right to win any money placed in the pot after the draw. (In other words, a "seven" may not be checked unless the player intends to fold when another player bets). Another common rule in low-limit games is that a player who checks on the second betting round may not subsequently raise on that round. This latter rule is never used in games with a pot limit or no limit betting structure.