I am generally against what I call "chasing the river". This happens when someone has a straight-draw, inside-straight-draw, flush draw, and even and open-ended straight-draw, and keeps calling bets all the way to the river card. The odds just aren't usually there, but people do it all the time. You're sitting there with just the river card to go and you need one card to make your straight - let's say you need a 10. There are only four 10's in a deck. 4 out of 52 cards is only 7.7%. Another way to look at it is that by chasing a river 10, you have a 92.3% chance of missing, and maybe it's worse than that because someone else already folded a 10.
Maybe your chasing a flush on the river. In this case, you have 4 cards to a flush and you need just 1 more. 4 of the 13 cards of that suit are already accounted for, so just 9 possible cards remain that could help you make that flush. 9 out of 52 cards is 17.3%. That means that you've got a 82.7% chance of missing your flush IF nobody else already folded cards of that same suit and IF your remaining opponents aren't holding any. Those are big IF's.
Maybe you have an open-ended straight-draw where you could make a straight with a card on either end of the 4 consecutive cards you already have (including the board). Let's say you have 8, 9, 10, J. In this case, you could make a straight with either a 7 or a Q. So, you chase the river. Your odds are going to be a little less than the flush draw above since there are 8 possible cards in the deck that cold make your straight. 8 out of 52 cards is 15.4% . As long as some of those cards weren't already folded or someone else isn't still holding them, you have an 84.6% chance of missing.
When you hit your cards, your thrilled and your opponents cuss you, but usually you miss and you donate. The odds just aren't in your favor to chase the river. As with many other aspects of poker, there are exceptions to not chasing the river. I'm not getting into those exceptions here. For now, just understand that the odds are against you and your buddies at the table are going to cuss you when you hit.