An adversary system is the opposite of an inquisitorial system in legal terms. In the latter case, when there's a dispute between two civil parties which has been brought to court, the dispute is analyzed and settled by a judge, or a team of judges. In contrast to that, an adversary system allows each side to represent their own side of the argument, usually by using an attorney who represents the individual and lays out their own arguments as to why the case should be ruled in their client's favor - and the judges simply make a decision based on the presented facts.
The adversary system is almost universally used in courts across the world nowadays, and it remains the most popular way of solving legal disputes. This does not completely exclude the inquisitorial system from play though, as some special cases still rqeuire it to be applied - though these are always the exception and not the norm.
The judge rarely takes a decision on their own in an adversary system, and are most commonly assisted (or completely replaced in this duty) by a jury, which listens to the arguments presented by both sides and makes an informed decision on which side should be granted victory in this particular cases. In cases where the jury makes the decision entirely, the judge is still required to approve and announce it to the parties involved in the lawsuit. The adversary system is commonly seen as more objective and rightful than the inquisitorial one.