The board is a sentence, the river is the period (or explanation mark!). To get here you've had a good draw (and don't kid yourself every hand you had on the flop and the turn has been a draw. If you flop a set of kings you are on a draw to a full house--if you don't make that draw you may and often are in trouble. Even when you flop a made hand like a flush you are still drawing, hoping that the board does not pair).

What this means is that all the big betting and raising has already been done. If you didn't do it on the first three betting rounds don't expect to be able to do a lot of it here unless you are beaten (or quartered).

It also means if the pot is 5 handed and you had the nut flush on the turn and the board pairs and there is action on the river that you are beaten. Fold it.
When you have the nut low and a pair...

If you have the best possible (nut) low hand and a pair then you want to be in there raising to drive out marginal (but better) high hands, especially if someone has another low. Doing this can be the difference between getting 3/4 or 1/4 of a pot and many people won't do it for fear that they will be quartered. Well sometimes you will be quartered but you only lose a tiny amount of money for the hand when you are and you gain a huge amount when you get 3/4 or scoop the whole pot.
When you have the nut low and no pair...

If you have no chance at all of winning high then you want to finish the river for as few bets as possible. In almost every conceivable case you'll want to call the final bet or bets to get your share of the pot even if it is to reduce the amount you are losing but don't bother raising with this hand unless you are reasonably sure that you are facing two or more other high hands based on the betting of previous rounds.
When you have the nut high...

How you play this hand on the river depends on a couple of things. Certainly if you have the highest possible high hand you want to get money into the pot. The possible exception would be when your high hand is a straight and you believe that someone else may have the same straight. For example if you had an ace high straight on the turn and capped it with two other players raising you and then a third low card falls on the river then there is a decent chance you've been quartered with the high. In this situation you want to play the hand as you would a nut low with no high.

If you do have the single highest hand then don't be afraid to bet it out on the end. Many people make the mistake of checking the nut high hoping for a check raise on the end or that they will scare out other hands but especially if there is any low possible you should bet out and let the lows battle it out for their half of the pot.
When you have the nut nut...

Occasionally you will find yourself with the nut low and the nut high simultaneously. The most common way this happens is when you have A2 with a suited ace and you make your flush and low.