River play

After the river card is dealt, there are no more possible improvements. Again you have to ask yourself if the river card has helped one of your opponents, and if you still hold the winning hand. The river card may have helped you, but do not get carried away: it may have helped your opponent even more, so again you must also think about what the other guy might have.
Normally, if you have gone to the river and there is even a small chance that your hand is winning, you don’t want to fold to save one bet and loose a pot that you would otherwise have won. Let’s assume you have to pay 1 $ to have a look at the other guys hand, and the pot is 10 $. You only have to win the pot 2 times out of 10 to have a positive expectation (Investment: 10$ for ten calls, versus 2 won pots with a win of 20 $).
Vice versa, when the pot is very big, do not expect that your opponents will fold to a bluff: the high pot odds will justify their calls even with weak holdings. They may fold if it’s 1 $ to them to win a 4 $ pot, though.
There is one special situation that occurs rather often: you are last to speak, your opponent checks to you and you are not sure if you are holding the winning hand. Now you can bet, and if you were ahead, the other guy will fold most of the time – you have not won an extra bet. Or you can bet, and the other guy calls or even raises, which most of the time means he has the better hand – you have lost an extra bet or two. So here you are in a no-win situation. The right thing to do would be to check, too, and take the free showdown.