The notion of reopening the betting refers to whether a player may raise after another player has gone all-in for less than a full bet. The general rule varies between limit and big bet games.
No Limit/Pot Limit Edit
Bob Ciaffone drafted a set of comprehensive poker rules. Here's what he has to say about reopening the betting for no limit and pot limit games:
"All raises must be equal to or greater than the size of the previous bet or raise on that betting round, except for an all-in wager. A player who has already checked or called may not subsequently raise an all-in bet that is less than the full size of the last bet or raise."
That's fairly clear. Consider this action from a no limit Texas hold'em game:
Three players have seen the flop. Player 1 bets $100. Player 2 calls, and Player 3 goes all in for $175. The other players, who have already acted may only call the remaining $75. Betting will begin again on the turn with the money going into a side pot.
Now, in the same scenario, Player 1 bets $100. Player 2 calls $100. Player 3 goes all in for $200. Since $100 is a full raise, the betting is reopened and Player 1 or Player 2 may raise. The extra money will go into a side pot.
House Rules Edit
Remember that there are always going to be rooms where this rule is interpreted differently, so don't be surprised if you find local variance.