A player is said to be taking a betting action in the dark if they take that action before seeing the card(s) that are dealt just prior to the betting round. The most common uses of this phrase are to check in the dark and to bet in the dark, most often in either Hold 'em or Omaha. Since taking an action in the dark is an unusual situation, it generally cannot be done via hand signals (e.g. tapping the table for checking) and must be done verbally. In some casinos, even putting chips out to indicate you are betting in the dark may not be binding, although in most casinos the act of placing chips beyond the commitment line would be binding.
The phrase comes from the idea of betting without sight, as though there were no light in the room and you were unable to see the cards. Once the cards are exposed, you then have "light" and can see the cards to bet on.
A player can only act in the dark if they are first to act on the forthcoming betting rounds. Once the first player has acted in the dark, the second player may also choose to act in the dark, at which point the third player may also choose to do so, and so on. But a player may not act in the dark if they follow a player who has not acted in the dark.
Dark actions are generally considered binding in casinos, though some casinos may have house rules that state that dark actions are not allowed, in which case stating you are taking an action in the dark would not be binding.
Note that taking action in the dark does not convey any special privileges (unlike, say straddles, which allow a player to "raise themselves"). As far as game play is concerned, acting in the dark is treated identically to simply acting in turn after the exposure of the card(s).
The only benefit to acting in the dark is to convey to your opponents the idea that the forthcoming card(s) will not affect the strength of your hand. If you "check in the dark", you are telling opponents that regardless of the next card(s), your hand is not worth betting. If you "bet in the dark", you are telling opponents that regardless of the next card(s), your hand is worth betting. Note that your telling your opponents this does not mean it must be true; you can use a "bet in the dark" strategy to bluff opponents, by indicating you have a strong hand when you have a weak one instead; or vice versa.
Some players confuse the phrase "in the dark" with the phrase "in the blind", which means something entirely different. Still, you will often hear inexperienced players say just prior to the flop "I check in the blind". While they may indeed be in the blind, they are intending to check "in the dark".
Check in the dark Edit
A player may choose to state that he "checks in the dark" if he is the first to act on a betting round (or he is the first to act on a betting round after other players who have all also checked in the dark), as long as he does so prior to the card(s) for the betting round being exposed. Once a player has "checked in the dark", he is not allowed to change their mind after the card(s) have been exposed; action continues to the player's left as though he had checked normally in turn.
This is raw content, possibly from a cross-post licensed by its author, that needs to be cleaned up.
The idea of this check is that it carries an implicit threat of a check-raise. You're transferring last action to your opponent, so to speak, by forcing her to act without any meaningful input from you in the present betting round.
The flaw, of course, is that if your in-position opponent checks behind, the dealer doesn't ask you what you'd like to do on this round. He merely raps the table, burns and deals out the next card, or if it's the river asks the players to show down. So the price of possibly scaring your opponent from betting (because she can't tell whether you've made your hand) is giving a free card or showdown when you might not wish to have done so if you'd seen the card.
Since you can always check (and threaten a check-raise) after looking at the next card, a check in the dark is only useful against players who are very psychologically vulnerable to it. 
Bet in the dark Edit
A player may choose to state that they "bet in the dark" if they are the first to act on a betting round (or they are the first to act on a betting round after other players who have all also checked in the dark), as long as they do so prior to the card(s) for the betting round being exposed. They must place the chips that would be required for the bet onto the table, beyond the commitment line, in order for the bet to be valid. Once a player has "bet in the dark", they are not allowed to change their mind after the card(s) have been exposed; action continues to the player's left as though they had bet normally in turn.
If a player does not state that they are betting in the dark, but does place the correct number of chips onto the table prior to the card(s) being exposed, in most casinos this still counts as a "bet in the dark" and is binding. Players should announce their action, however, as a courtesy to the dealer and the other players.