One example is the use of a voting system to choose candidates to hold political and military power. This is adversarial since it requires each candidate to convince voters that they are more trustworthy in the expected future circumstances, than their opponent.
Another example is the use of a criminal court or civil court to decide the social attitude to an alleged wrong-doing of a defendant, and penalties to be assessed, and restitution to be awarded to their deemed victim. This is adversarial as the opposing attorneys are competing to convince the judge to include or exclude evidence or witnesses, and competing to convince the judge or jury of the guilt or innocence of the defendant, and severity of the impact of the actions (if guilty) on the plaintiff or victim. Lawyers must be held to rather specific ethical codes, e.g. rules of civil procedure, in order to ensure that their tactics do not cause an undue burden on larger society, e.g. freeing defendants who have admitted that they are not only guilty but intend to offend again. Lawyers differ on whether the process should be seen as strictly adversarial, in order to ensure they retain the trust of clients and the overall process retains the trust of society, or whether the ethics of the larger society should play a role in their behavior, e.g. freeing O.J.
A third example of an adversarial process is the operation of market systems, e.g. commodity markets. In these, bid and ask prices are constantly compared, with sellers representing goods as being valuable and buyers haggling and claiming they are less valuable. Product markets tend to focus on the comparison of sellers' products with other sellers' products - the adversarial process itself making trustworthy information, e.g. as published in Consumer Reports or the Better Business Bureau, hard to compile and to obtain. Health advocates often claim that market systems are very difficult to reconcile with food, nutrition, agriculture or medicine's need to work well with living systems - a key complaint of the anti-globalization movement.
Despite these hesitations, adversarial process is obviously embedded deeply in Western civilization, and it's difficult to imagine working without them. The alternatives, including consensus decision making and deliberative democracy (which tend to alternate adversary, discussion, and voting over a longer period of time allocated to make the decision and explore implications), are ancient. However, they are less well studied in political science and economics. Most commentators are suspicious of the utopian goals of the advocates of such adversarial processes, viewing competition as essential to a good result.
Other activists argue that adversarial process works well and should probably be expanded to areas that are presently less adversarial. For instance they advocate "science courts" and "prediction markets" that would force the scientist, economist and technologist to put reputations and money on the line, rather than trusting them based on reputation without a disciplined follow-up to see if they were right or wrong. These ideas are increasingly popular in part because such alternative courts and markets can be easily started up on the Internet.
Some view the wikipedia as an adversarial process, since alternating edits to an article are often seeking to balance what is seen (by each author in turn) as an extreme point of view. The idea of wikipedia:neutral point of view inherently assumes that such adversaries will eventually converge to some kind of agreement, more or less the same assumption as is made in supply-demand curves in economics. However, like economics, an underlying political economy wherein influence and power is traded, limited access to tools, fears of censorship or the need simply to allocate more of one's time to 'the real world', probably have just as much to do with where a dispute ends, as does any kind of 'balance'.
In this respect, wikipedia is no different from political, economic, or court competition: those who have time and skill and friends and resources tend to prevail. This remains the most fundamental objection to adversarial process, and so most efforts to improve it, e.g. electoral reform, try to balance the resources available to both sides to make their case, and provide maximum opportunity for unpopular but valid views to make their way into the forum to be considered. In general, economic processes do this better than any other, as new information is incorporated in market prices often within minutes of some important event - due to the adversarial process of buyers and sellers - but also due to the consensus on credit, clearing and pricing systems, without which the entire market would simply dissolve into a puddle.
See also: consensus decision making, voting system, advocacy