I think my style on the river will differ in a cash game v. a tournament.

In a tournament it is pretty simple. If I'm out of position and took the lead on the flop and get called down or re-raised on the flop, I as myself on the river- "If I bet and get reraised or opponent pushes, am I prepared to call?"
The answer will depend on the action and any read you have on the opponent but if the answer is "yes" then bet the river. If the answer is "no" check.

In a cash game, you can be a little more liberal on the river. Again, it depends if you have a read on the other guy, how many people are left on the river and what is the texture of the board. Did a draw get there?
Sometimes on the river you can bluff if the draw hits, it takes balls and you have to have a good feeling that the other guy will fold. But again, if you are relatively weak such as top pair and good kicker and you can't shake your opponent I would ask the same question as above when deciding how to act out of position on the river.

If you are in position on the river and it is checked to you and you are only playing top pair I would check and showdown since you don't want to offer your opponent a chance to bluff you with a re-raise on the river. I would note that not many players have the balls to pull that off especially online at lower limits since they will usually get called anyway.