Texas Holdem Betting at the river
One thing you will notice if you have gotten this far with your hand (and many many times you won't) is that you have a lot of company. This is also where you will suffer through a seemingly endless series of bad beats.
You should plan on playing much more timidly here, especially if any scare card has fallen. Your pair of kings with an Ace kicker was good on the flop but if you did not improve beyond that then do not be surprised to be beaten by some of the strangest two pair (and beyond) that you've ever seen.
Also current behavior will be to raise you if your opponent did make 2 pair on the river. A check will show weakness and someone with a king and a worse kicker will often bet for you and you will save yourself calling an extra bet by the two pair river chasers.
Try to see all the possible straights, flushes, etc. on the board. Consciously do that because when you are focused on your hand it is sometimes very easy to miss a straight or flush that just appeared. Especially if you are focused on your opponent having a similar (but smaller) hand than your own. Be especially careful if you make a straight and a third flush card has also fallen even if that third flush card was a backdoor flush.
The bottom line is that it is often worth checking and calling on the river if you have a good but not great hand. Save the aggression for when you have the nuts or near the nuts.
If you've followed the advice here you'll be getting to the river with at least fairly strong hands (and many times with the absolute highest hands [the nuts!]). You should almost always call a single bet on the river if you have any reasonable hand at all.
It can be a huge mathematical mistake to fold on the river if you are up against one or two opponents. Most of your folding should be before the flop, on the flop and on the turn. If you paid for the turn you should call with any reasonably strong hand.
The exception to this is where there are multiple callers to a bet on the flop. Your odds of winning with an average hand go down exponentially the more callers there are. Calling a bet after more than one person has already called is called "overcalling" and you need more strength to call each overcall--and if you have enough strength to be the fourth overcaller you should consider raising.
Don't get too upset if you lose a made hand (AA, top two pair, etc.) on the river. If you got your money in during the hand with the best of it that's all that really matters. In the long run you make money by making good decisions and in the short run you are largely at the mercy of luck. Keep making the right decisions! The rest will take care of itself.