In a cash game (let me repeat that, not in tournaments), once the players are all-in, they may decide to deal the board cards more than once. They then balance the results.
So, if I were to go all-in on a flush draw while you were sitting on a set, and we "ran it twice" we would deal the turn and the river. Whoever won would get half the pot. We would then redeal the turn and the river. Whoever won would win the other half of the pot.
Here's a video example from High Stakes Poker featuring Daniel Negreanu and Jennifer Harmon.
Why run it twice?
It reduces the variance. At the lowers stakes that doesn't seem like much of an issue. But if you have say, $60,000+ of your own money invested in a hand, you might want to reduce the blow you'd take if someone sucks out on you by running it twice.
It's important to note that your Expectation doesn't go down by running it twice. If you're a 85/15 favorite, you'll still win 85% of the time over the long run. It's just that when someone does manage to hit one of their outs, and you lose the pot, you'll have another chance in the same situation right away.
Where do they run is twice at an online poker room?
Well, at least at the moment, the picking are pretty slim for this one. BUT, there is one network of rooms where the software allows players to choose to run it twice.
Carbon Poker, and the rooms on the Merge Poker Network have tables where players have the option to run it twice. They actually refer to it as "Deal it Twice".
Here are the conditions that the room sets out for Dealing it Twice:
Deal It Twice is only available on real money No-Limit Hold'em ring games.
Deal It Twice can only be chosen when exactly two players are all-in, with no other players active in the pot.
It is only offered as an option after two players have pushed all-in pre-flop or post-flop. Even if the players are in pre-flop, only the turn and river cards are dealt twice.
Deal It Twice will be available only on certain tables at specific stakes. Simply check your online poker client and browse our real money Texas Hold'em ring games to see which tables have Deal It Twice as an option.