For example, "Ice cream tastes good" is an offensive argument, while "Ice cream doesn't taste THAT good" or "There's no reason why ice cream tastes good!" is a defensive argument.
At the end of the debate, the judge must make a decision between the advocacies of two teams. Offense is the way that teams definitively differentiate between the value of their advocacies so that the judge can make an informed choice. Debate is impossible without offense; a debate between someone who said "ice cream isn't perfect" and someone who said "ice cream isn't the worst food ever" would be inconclusive because neither argument actively provides direction in choosing whether or not to get ice cream.
In a situation in which one team has offensive arguments supporting an advocacy and the other team only has defensive arguments against it, the team with offensive arguments will often win. Teams often use the phrase "risk of a link" or "risk of offense" to describe this situation so that the judge can immediately identify the sole source of direction in making her or his decision. Generally, the tendency for judges to err on the side of voting for offense has been called an "offense/defense paradigm," and most judges use it for lack of a better metaframework for analyzing arguments.
For example, if the 2NR goes for a politics disadvantage with a very tenuous link, the affirmative's natural response might be: "There's no way that our plan would cause a big enough disruption in day-to-day politics to cause your impact scenario." However, if the affirmative was winning no offense elsewhere and that argument was their only response, the negative would probably win because the negative demonstrated the only possible causal connection with which the judge could pick the best outcome.
The affirmative's best option here, of course, is to propose a link turn or impact turn, so that it can generate offense against the negative, or to argue that case impacts outweigh (meaning that the the 2NR's scenario is very unlikely, and that their case scenario goes conceded and has a bigger impact, so you vote aff).