Professional players often offer draws in chess and accept them. While draws can be very common in professional play, they appear to be very rare in casual play. Naturally, if you're winning, accepting or offering a draw seems silly, but what if there isn't a clear winner? When is a good time to offer a draw, and when should I accept one?
In casual play, the main reason to offer a draw is if the game is going on forever and shows no sign of concluding soon. Consider the motivation for the game. Maybe you're playing a game for fun after dinner, but at some point your wife is going to start making loud yawning noises. Maybe you're playing a grudge match with a friend, but both of you are hopeless at rook endings - which is where you've ended up. Maybe you made the mistake of starting a game with someone who takes hours to make any move. I've been in all of these situations! In these cases, achieving the win is secondary to the experience of playing the game. As soon as the experience becomes compromised, or real life starts to intrude, an escape becomes appealing.
One player may actually have an advantage, but if it's going to be necessary to grind out a difficult endgame to convert that advantage into a point, then it may not be considered worth it by the players to continue. A couple of times I have even conceded casual games that I believed I was winning, simply because I wasn't willing to continue to invest the time.
Another reason is if the players are amateurish enough that forcing a win is difficult or impossible. I once spent a frustrating afternoon watching two friends with very basic chess playing abilities circle uselessly for several hours, trying to figure out how to achieve a checkmate. It was like watching Brownian motion on a chessboard; essentially random checks with the vague hope of accidental checkmate. In the end, they gave up and declared a draw. This situation is depressingly common among the large group of casual players who possess no knowledge beyond how to move the pieces.
A final, rather dishonest reason is to protect the ego of your opponent. Too many casual players take losing at chess as a personal failure, or worse, an insult or demonstration of superiority by their opponent. If your opponent is your boss or your grandfather, and they take losing very badly, maybe a draw is a more political decision. In this case, the game is just part of a larger game - the game of maintaining workable relationships in the real world.
I genuinely can't think of a better answer than "when you consider there is no possibility of either player winning the game". If your opponent doesn't see this, but you can explain to them how a win for anyone has become impossible, so much the better for your ego.
Either that or if the last bus leaves in five minutes!
In general whether playing a "serious" (i.e. tournament, rated game) or playing casually a draw should be considered when a game is no longer interesting but isn't a clear victory by one side (which is when in either tournament or casual play the losing side should realize this and probably should resign).
What do I mean by "no longer interesting" - this will vary by your level of skill but it isn't uncommon at all for players a high level to try out an opening variation but if via their opponents play the game transposes into a position known from past games and openings both players may decide that they should call it a draw and in a casual situation play another game.
In casual play there are many types of games people play - casual games without a clock can take almost literally any amount of time both players want to dedicate to it (if play by mail this could be months or even years) but in person it is polite to play reasonably quickly and to offer (and accept) a draw when the game bogs down or shifts into a fairly known area especially if the goal of playing is to explore new avenues of play between players of nearly equal strength.
In casual games with a chess clock (which could be speed chess or could just be close to a tournament speed) again draws may be offered when the game is falling into a duller, more repetitive pattern - but the opportunity to win via time often means that one player or the other may have a real advantage and often presses that even in an otherwise drawlike situation.
I used to be a very serious chess player - my favorite non-accepted draw came in a team tournament, the last of the season when I was in high school. I offered a draw to my opponent even though I was slightly down in pieces (but had some advantages in position) because I knew that my team's position would not be effected by a draw or my win and I knew that my opponent would advance to the end of the year individual tournament if he accepted my draw.
However he either forgot that or decided to press his luck and he turned my draw offer down. But then proceeded to fall into time pressure and in a few short moves he lost his queen and I won the game handily (though on later analysis with correct play he should have beaten me at the time I offered him the draw - but in time pressure he was unlikely to have found the right set of moves to do so as the position was highly dynamic)
Offering a draw and NOT playing a game to the very bitter end is also a mark of respect to the other player - whether in a tournament or in casual play. Knowing when a game is drawn and equally importantly knowing when you have likely lost the game is a sign of a chess player of some skill.
One of the signs of chess players of less skill tends to be that they play every game to the bitter (and often for more serious chessplayers observing painfully slow) end. This isn't to say that you should resign or should offer (or accept) a draw when down a bit or when in a seemingly disadvantageous situation but it is to say that you should recognize when the game is mostly equally balanced and rather than play for dozens or more moves and often hours more, take (or offer) a draw (or if losing resign) and play another game. You will learn far more in the long run when you learn to recognize the difference between games which still have a lot of dynamic possibilities and games where they are drawn or where one side has a clear path to victory (even if it isn't a forced checkmate).
I disagree, strongly, with the other answer that suggests the only times when a chess game should be drawn is when it is technically a forced draw (i.e. neither side has sufficient pieces for a mate).
Above all else chess when played casually should be fun. Playing games out to their bitter end is rarely fun for both players and should be avoided.
Usually, when chess in analyzed, it's done with the benefits of a decision judged solely on the game state. That is, players are modeled as caring only about winning the game. But of course people are motivated by more than that. Deciding on whether to accept a draw depends on factors external to the actual game. There are many overlapping motivations for playing chess: money (at the high end), ego/bragging rights, fun, developing one's skills, bonding with someone else, etc. When to seek/accept a draw is going to depend not only on the board state but on what your motivations are.
For example, if your motivation is to practice chess and increase your skill, there can be value in playing on from a position that experts would consider drawn, if you haven't fully mastered how to maintain the draw from that position. On the other hand, you might want to accept a draw from a position that you think you could win, if you don't think that executing that win would develop your skills at all.
If your motivation is fun, then the question is whether the game is still fun. If all that is left is the application of endgame theory that you find tedious, then a draw would make sense. But if you would enjoy seeing how the game would play out even if it ends up a draw, then it would make sense to not offer a draw. As for whether you should accept a draw, the fact that your opponent is offering one suggests that perhaps they aren't having fun anymore, so the very fact that one is being offered is an argument for accepting it.
In chess draws are not usually arbitrarily decided. There are very specific conditions that trigger draws. The most common is just the kings (and a knight or bishop) remaining. This is the only time a draw should be accepted. Any situation that is not listed as a read draw should be played out. Chess games cannot be stuck if all rules are followed including the automatic draw for repeating a situation on the board, and people knowing basic mates (like the queen king mate)
Why do they happen often in high level play? In chess the black side is considered at a disadvantage and in a very high level game is playing for the draw. So the goal of a grand master is often, win all white games and draw all black games.