Definition of paired-board
A paired-board is when a table is paired. For example, if a board is 2-5-6-7 and the river brings in a 2, we will say there is a paired-board because the board is "paired" (there is a pair on the board). The reason for taking the paired-boards into account is that on a board with a paired-board, there is now a possibility of a full hand. Just as you have to be wary of a third spade on the river when you have a straight (your opponent might have the flush), you have to be wary of a paired-board when you have a flush. (our opponent could have a full hand).
The paired-boards to Hold’em and Omaha
A paired-board changes the dynamics of a table, but at Omaha, much more than at Hold’em. Hands full on Omaha will be at least twice as frequent as on Hold’em. Some paired-boards are more to watch than others. For example, if we go back to our table presented above, on Hold’em, our opponent will rarely have a square of 2 and probably a full hand with only 88 and 99. Few players play 42 or 92. On the other hand, on an A-K-J-Q-A board, the ace will be a more worrisome paired-board if you have a flush or straight since AK, AQ and AJ are hands that your opponent is more likely to have than a 4-2 as the highest.