Draws have different meanings in poker, depending on the context.
A Stage of a Hand When Players Draw Cards[]
The draw in a game which features draws (such as classic Five-card draw, or Badugi) is generally held after the inital betting round. In it, players take turns (following the same order as betting in the previous betting round) discarding cards from their hands and replacing them with new cards being dealt to them from the deck. Some variants have specific rules about how many cards players are allowed to discard and draw.
In general, draws should be done by the player removing cards from their hand and passing them face down to the dealer, who will count them, deal the player that many new cards from the remainder of the deck, and then muck the player's discarded cards. Draws are nearly always declared out loud, as in "I'll take 2" or "I'll draw 1 card".
Players who choose not to draw any cards during the draw are said to be standing pat. This indicates the player holds a very strong hand which is unlikely to be improved by exchanging any of its cards. In five-card draw, for example, standing pat implies the player has a full five-card made hand, such as a straight or a flush. This strategy can also be used to bluff other players.
The number of cards a player chooses to draw is significant information that other players can use to deduce the strength of a player's hand. Accordingly, some players have devised bluffing strategies for drawing cards by altering the number of cards they draw to imply different things about their current hand strength.
A Partially-Completed Hand[]
A draw is a possible card that could be dealt to you (or instead, to the community cards, in a game which uses them) which, if it is dealt from the deck, would cause you to make your hand and end up with a very powerful poker hand.
Having a hand with a draw (or, as is often the case, more than one draw) implies that your current hand is very low in value, perhaps even only a High Card or a low pair. If you already have a powerful hand and also have a draw to a more powerful hand (e.g. you have three of a kind, but are drawing for a full house), this is typically called a Redraw.
The hands most commonly to be called draws are straight draws and flush draws, where a player needs only one more card to complete a straight or a flush. If the player needs a card of only one specific rank to complete their straight, it is called a gutshot. If they can complete their straight with a card of either of two different ranks, the draw is called either an open-ended straight draw or a double bellybuster, depending on the orientation of the cards they hold. Flush draws, by definition, can be made with any card of the suit in question. A straight flush draw is a hand become a straight flush - while straight flushes themselves are much harder to draw to though since they need specific ranks of specific suits, straight flush draws are considered quite valuable all the same since also by implication (and much more commonly) a straight flush draw is also drawing to both straights and flushes (assuming they are not so already). Also, in games that allow players to make hands from more than five cards, there are also many examples of hands that are straight and flush draws without being straight flush draw.
If a hand can be made only by receiving two consecutive draws (e.g. you hold two spades in your hand in Hold 'em, and the flop contains only one spade; you need two consecutive spades to make a flush), it is called a backdoor draw. Backdoor draws are, by definition, much less valuable than simple draws, since the odds of getting both consecutive cards is much less than merely getting one.
A dead draw is any draw that, if made, can still not be the best hand - either because any of the cards the player could use would make the same hand but with a higher value for an opponent (this can happen only in community card games), or because an opponent already has a better hand (e.g. a full house) than the hand the player is drawing to (this can happen in all games). Having a dead draw is not the same thing as drawing dead - for a player to be drawing dead means that there is no card in the deck that would give him the best hand, on the other hand it is sometimes possible to have both dead and "live" draws - for example to have straight and flush draws where if the straight were made it would be bested by an opponent's higher straight or flush, but the if the flush were made it would be the best hand. For example consider the following hypothetical Hold 'em hand:
Alice: Kc Jc
Bob: Ac 6c
Board: 3c 9h Td 8c ?
Bob currently has the best hand (Ace high). Alice has a double gutshot straight draw and a flush draw. Bob has a gutshot straight draw and a flush draw. Bob's straight draw is a dead draw because if completed it would make a better straight for Alice. On the other hand, Alice's flush draw is a dead draw since if completed it would make the nut flush for Bob.
Good strategy in a game with community cards or upcards, like Hold 'em, is to always be aware not only of your own draws, but the potential draws your opponents may have. If the turn or river cards bring cards which could possibly complete a straight or flush draw, it is wise to consider the possibility that one or more of your opponents may have made their hand.