This is a complex subject about which people say many different things. Be warned that if you use negative-sum game in the same way that I do, many people will disagree with you in small but important ways.
A negative-sum game (NSG) is a situation in which either all participants lose utility, or in which the sum of all utility and disutility at the end of the 'game' is less than the total utility at the start of the game.
Except, we don't really have a fully standardized definition, and most people who use the phrase do not use it to apply specifically to utility, but to things like money and power. And even then, they often use these terms inconsistently. To make this clear, let's look at some examples.
Gambling:
Gambling, especially in a casino or through a bookie, is often cited as a negative-sum game. While this is a useful example in that it is one that most people understand and have spent some time thinking about, it is not usually a NSG.
The argument goes that over time gamblers tend to lose money, primarily because the house takes a cut. Since on average gamblers should assume that gambling is usually going to shrink their bankroll, they should view it as a NSG, and avoid playing. This assumes that the gamblers are gambling primarily for money and not for the fun of gambling, which in an ideal world would not be true -- we would hope that they are gambling precisely because they value the utility gained more than the money lost. It also assumes that the house is not part of the game, which is a framing that may be rhetorically useful in convincing someone not to gamble, but is hard to defend in other contexts.
A full analysis of the situation would be that gamblers pay the house/bookie a small sum in return for providing them with the utility gained through playing games of chance; the rest of the money is distributed with varying degrees of randomness among those people who enjoy gambling, resulting in a zero-sum game moneywise, and a net gain in utility (i.e., a positive-sum game), as gambling is primarily engaged in by those who enjoy gambling.
This is complicated by cases of people addicted to gambling, for whom it is appropriate to judge utility differently, and the cases of people who need money so badly that it is worth risking a small amount to get a chance at a larger amount, whose judgement we tend to distrust. It is worth noting that in both of these cases we are looking at a person and saying that they are misunderstanding their own best interests; this is something that basic game theory does not attempt to accommodate for, and goes beyond the basic analysis of negative-sum games.
Arson:
Arson is nearly always a NSG, as are most crimes. Generally speaking, when an arsonist makes the judgement that they will be happier if your house is burnt down, they do not compare their gain to your loss, and believe that it will be a net gain overall; and for those morally good arsons who do run this calculation, I suspect that they are still wrong more often than not.
The case against arson is stronger than simply suspecting arsonists of having unethical mindsets and/or bad judgement in setting their fires. Society works hard to make crimes a NSG, and even if you correctly judge that any given felony benefits all involved on average, the police will work to try to make things worse and rebalance the equation. The government is playing a bigger game, where they try to make each given crime a zero-sum game or worse. Even if they fail in punishing the criminal enough to bring total utility down to a zero-sum, the increased burden on the taxpayers probably drags each crime into NSG territory. The hope, of course, is that if each of these individual games are costly enough, people will stop playing them.
Variable-Sum Games:
Most of the games that are identified as NSGs are in fact variable-sum games; games in which the outcome may be positive and/or neutral and/or negative, depending on which strategies players use.
An example of a variable-sum game would be war, a type of game that is often accused of being a NSG (and often correctly). However, one can easily imagine a situation in which an oppressed population gains utility by overthrowing a small group of oppressors, resulting in a positive-sum outcome, i.e., increased overall utility. Or, alternatively, in which the oppressed population is unsuccessful in revolt, and is oppressed more cruely, resulting in a negative-sum outcome.
In contrast, gambling can be used as a good example of a fixed-sum game, if we consider only money and not utility. When you place a bet, you may lose your money, or win money, but the entire pool of money is fixed; some will go to you (or not), some will go to other players (or not), and some will go to the house. But no money is created or destroyed.
In many cases, statements that something is a negative sum game are actually intended to indicate that there is a high probability of a negative outcome, or, in many cases, the observation that now that the game is over a negative outcome happened. This is not useful in analysing the way that 'games' can be improved; condemning a system because one time it went wrong is the opposite of trying to fix the system. However, it is worth identifying when negative-sum outcomes are likely, as those often can be avoided, and avoiding them is generally equivalent to 'solving the game'.