You have the right to play on and it is advisable for you to do so. It is better to try and find some good moves when you are losing because you'll become more resourceful. There will be games where you are sure that you'll lose but you will not lose in the end. Just try to make some threats with material you have left.
The only thing you shouldn't do is stall on purpose when you are lost. For instance don't let your clock runs out in a losing position or something similar. Other than that, it is your time and you have the right to play on.
Being a good sport questions

@nklristic, thank you for the comments. No, I wouldn't ever waste someone's time purposely, though I could see perhaps where I'm thinking and thinking it might seem like that.
For the times when I'm playing and, for example, someone loses their queen and promptly quits (I had this happen on about move 10, pretty close to the beginning of the game) is that considered poor sportsmanship?
I did have one game where I lost MY queen quite early but kept on playing and eventually managed to win, to my great surprise. So those games are fun to have, and I wish people would keep playing even if I make a good move.
And again: Say we are 3 or so moves away from checkmate (my win), and the person quits...is that something I should feel upset by? Maybe it's just me but there's something very nice about that final "Checkmate" move and I feel a bit robbed when someone quits before that happens. Is the game rated differently if they do? Will their chess rating go up or down due to that decision rather than reaching "Checkmate"?
Again, thank you for the answers. I am new here and so far finding this a lot of fun. :-)

You're welcome.
They as well have the right to play on or resign. So you shouldn't feel upset. I understand the urge for you to want to play on till checkmate, but it is their right to resign if they want to do so.
As for the rating... it doesn't matter in which manner you've won the game. If you win, you will gain the same amount of points regardless of the manner in which you've won the game in question.
Yeah, I had a game where the opponent didn't find mate in 1 (I could avoid it previously by giving up a queen for his rook, but I would just lose more slowly). I managed to trick him by offering him the free rook that I didn't need, he took it, and after that I had a forced mate in 3 thanks to my queen, so I won. Those things happen, especially on lower levels.
@nklristic, thanks again! Great to know (and yes, I could see all kinds of reasons why someone might want to end the game -- including someone knocking at the door or an emergency -- so I won't take anything personally if they do that. And I'm definitely at the "lower levels" (sigh) so I'm sure I'll see a lot of this going forward! :-)

You should play on until you feel that there's nothing further to be learned by continuing.
One thing that is considered very bad sportsmanship is to deliberately stall and delay, refusing to move anything until the time is almost expired. This is a very different matter from just "playing on in a lost position", since deliberate stalling is a refusal to play any move, hoping that the opponent will become frustrated at looking at the non-moving chess board and will quit the game, giving the staller the win by default.

Yes so it's a bit rude to resign, but if you're out of the game still don't. You can learn from your mistakes.

(As an example, I never resign unless the opponent’s advantage is overwhelming, and according to the meta-analysis site I use, I win something like 45% of my games where I’m down a piece or more, which is twice the average at my rating. Not resigning and looking for opportunities to improve my situation certainly contribute to this.)

I've just joined the site recently and wondered the same things so thanks for the original post and the subsequent answers. For the record I am a complete beginner so what follows probably only applies to fellow beginners.
I agree that when someone resigns against you it is frustrating in that you haven't been able to try and beat them via checkmate - missing out on valuable learning experience. However, I have discovered you can continue the game against the computer from the point where your opponent resigned or maybe even any point before then (although I have to change the bot level as it seems to default to maximum). Although not as good as continuing against the human opponent this at least gives you the chance to learn skills that you wouldn't have been able to due to your opponent resigning. Apologies if you already know this.
When it comes to resigning yourself I try not to (although it is sometimes easier said than done). For example in one game I was not doing well so I just resigned. However, when I went through the analysis I realised there had been a position just a few moves before I resigned where I could have check mated in one - even although I had been well down in the game throughout. Due to my inexperience I never noticed this. Talk about frustrating!!!!! Also, sometimes I have found that even although you are getting well beaten your opponent finds it hard to check mate you and you may just salvage a stalemate or even a win on time (albeit psychologically that doesn't feel like a win to me even although it is recorded as one so I won't look a gift horse in the mouth). Although it may annoy your opponent they themselves have hopefully learned from it as well in that they won't repeat the mistakes they made.
As an aside I am not convinced I would have as many wins as I currently have, had my opponents continued against me as I am not the best at spotting checkmates etc - although to be fair I have only been playing the game for just under three months and can't believe how addictive it is.

As you have seen, at low levels blunders abound, so just dropping material that theory considers decisive doesn't mean you should resign. Finding unexpected resources in difficult positions is a skill worth cultivating. If the position gets so bad that even a gross blunder by your opponent will still leave you in a hopeless position it is probably best to say "Good game" and resign.
Also, no one likes it when you have blundered into a lost position but have a bit of a time advantage and just play as fast as you can, hoping your opponent won't be able to make enough moves to mate in the time he has left. (Of course if your opponent is in serious time trouble you are justified in playing on in a bad position under the "he got the advantage by taking more time for his moves and now he has to prove that he can win in the time he has left" theory, but again don't just fling out any move ASAP to run down his clock, but continue to play seriously.
If you are worried about being a bad sport here then you are unlikely to be capable of being a bad sport. The bar isn't all that high in internet chess. Don't be a jerk in chat and don't do things for the sole purpose of wasting your opponent's time out of spite.
That quitting is just fine in chess is a cultural thing that probably wouldn't exist if chess had always been played with short time controls.
Yes so it's a bit rude to resign, but if you're out of the game still don't. You can learn from your mistakes.
it is no way 'rude' to resign. a win is a win. take them any way you can get them.

Yes so it's a bit rude to resign, but if you're out of the game still don't. You can learn from your mistakes.
It is absolutely not rude to resign.
You could make an argument that it is rude to not resign, but there is no argument whatsoever to be made that resigning is rude.
Years ago I used to resign if I blundered a minor piece (edit: I still do most of the time). I remember I lost both central pawns in the opening once and resigned. I don't care about winning. I never have. I care about enjoying the game and improving at the game. What am I going to learn from playing on in such positions? How could I enjoy such a joke of a game?
Anytime you play on in a lost position purely for the purpose of learning, as you say you do, what you're doing is making a decision. You're deciding that the time you're going to spend on your current lost position is going to be more beneficial to your growth as a player, than spending the same amount of a time on a new game.
Everybody here has said that you should do what you feel is best for you. I agree. However, they've said that this might be to play on in hopeless positions. With this, I cannot agree. You should of course do whatever you feel is best. I just personally don't see how playing on just to see how you're check mated, could possibly be the best use of your time.

However, they've said that this might be to play on in hopeless positions. With this, I cannot agree. You should of course do whatever you feel is best. I just personally don't see how playing on just to see how you're check mated, could possibly be the best use of your time.
Particular in relation to beginners, I think you're on the wrong track with this, for two reasons. First is that a beginner is more likely to feel a position is hopeless than perhaps it actually is. More than a few times I've blundered a queen and gone on to win. Often, the other player lets down their guard and I make up the material deficit. Sometimes they don't know what to do in the endgame, I promote a pawn, and use my new queen to clean up their remaining pieces.
I'm a relatively weak player, but chess.com's matchmaking is pretty good at making sure I play other people who are capable of the same errors I make. Down a queen? Certainly play on.
Second, for every game I'm down material, there will be one where I'm up, and seeing how my opponent exploits that advantage can provide good ideas for myself when the tables are turned.
The only point where I definitely resign is: Lots of time on the clock, my opponent has a clear path to victory, and they're making moves that indicate that they know how to win and are heading right for it.

I happy to agree to disagree with you, Paleobotanical.
I believe you're focused on whether or not the game can be won, rather than improving your game. That is entirely up to you. I would never say that a beginner can't win from being a Queen down. I just don't see why it matters if they can.

Only king against an army could be a good stalemate practise. Don't capture anything just move around, hide behind them. Try to sharpen your stalemating skills That's all there is to it to learn.
Other than that, playing a lost position is usually a waste of time. But still you have right to play. Unless you stall the game. a normal player should checkmate you quickly already.

I happy to agree to disagree with you, Paleobotanical.
I believe you're focused on whether or not the game can be won, rather than improving your game. That is entirely up to you. I would never say that a beginner can't win from being a Queen down. I just don't see why it matters if they can.
Attempting to win from behind is a key test of tactics, which is exactly the skill that a sub-1000 player like me needs to practice most in order to rid oneself of the blunders that make one a sub-1000 player (and which likely led to being behind in the first place.) So, I don't see how you can say that it's not an improvement-focused choice.
(It also adds the benefit of seeing more endgames in the long run, which is itself good practice of a different set of skills.)

Attempting to win from behind is not a key test of tactics. It is a test, but what isn't? If you want to remove blunders from your games, surely focusing your energy on games in which a blunder is still possible, is the most basic of requirements? To lose a Rook for no compensation, when you were already down a Queen, is not a blunder.
I am a beginner and curious what the best "sportsmanship" is in the following situations so I don't upset anyone.
1. Is it "sportsmanlike" to keep playing the game out to its bitter end, even if I'm sure I'll lose? For me, it's sometimes educational and I'm learning, but am I wasting the other person's time?
2. What should I do to be a good sport if I know I just made a blunder that will cost me the game?
My reason for asking is that I've always played games basically to the bitter end, but am finding in playing online there are a lot of games that people say "You Won" (resigning?) when they've lost the queen or when I've made a good move...which (sort of) robs me of the chance to learn how to checkmate someone. Is there a way to indicate that I'd really like to play to the bitter end without upsetting someone?
I don't know whether that's considered gauche...can see both ways: One, you might want to skip onward to a game you've got a chance of winning, but on the other hand, maybe in doing that you're saying the (winning) person isn't worth your time.
Are there different views about this? I've been saying "Sportsmanlike" at the end of each game no matter what, and usually with a "Good game" autocomment, too. But I'd love to know more tips on how I should properly acknowledge these things and be a good sport, a good learner, and give people good games without draining their patience or time. Any tips and suggestions would be welcome! Thank you in advance!