The term is indeed an oxymoron. Libertarianism, as expounded in John Stuart Mill's On Liberty, is the doctrine that government should confine its interventions in the private sector to what Mill called "other-regarding" acts, which is to say acts that cause harm to nonconsenting strangers, as distinct from "self-regarding" acts, which are acts that harm only oneself or people with whom one has consensual relations authorizing acts that may result in harm. So, for example, if you are hurt in a boxing match, that is a "self-regarding" event with which the government has no proper business, provided the boxer who hurt you was in compliance with rules--to which you had consented--governing the match, and provided you were of sound mind and so could give meaningful consent.
Paternalism is the opposite. It is the idea that someone else knows better than you do what is good for you, and therefore he should be free to interfere with your self-regarding acts. Paternalism makes perfectly good sense when the "pater" is indeed a father or other parent and the individual whose self-regarding acts are in issue is a child. In its more common sense, "paternalism" refers to governmental interference with the self-regarding acts of mentally competent adults, and so understood it is indeed the opposite of libertarianism. The yoking of the two in the oxymoron "libertarian paternalism" is an effort to soften the negative connotation of paternalism with the positive connotation of libertarianism.
I would further limit the term "paternalism" to situations in which the government wishes to override the informed preferences of competent adults. The dangers of smoking are well known; indeed, they tend to be exaggerated--including by smokers. (The increased risk of lung cancer from smoking is smaller than most people believe.) Interventions designed to prevent smoking, unless motivated by concern with the effect of smoking on nonsmokers (ambient smoke, which is not much of a health hazard but is an annoyance to nonsmokers), are paternalistic in the sense in which I am using the term.
Thus I was not defending paternalism when I defended the ban on trans fats in New York City restaurants. If people are aware of the dangers of trans fats but wish to consume them anyway, the only nonpaternalistic ground for intervention, which I would be inclined to think insufficient by itself, is that they may be shifting some of the costs of their medical treatment for heart disease to taxpayers who forgo consumption of trans fats. If, however, people don't know the dangers of trans fats and it would not be feasible for them to learn those dangers (prohibitive transaction costs), and if as I believe the dangers clearly exceed any benefits from trans fats compared to substitute ingredients, then the ban can be defended on nonpaternalistic grounds, as I attempted to do. Another way to put this is that it is not paternalistic to delegate a certain amount of decision making to the government. There are some goods that government can produce at lower cost than the private sector, and among these is the banning of trans fats from food served in restaurants.
It might seem that the good could be produced just by competition-impelled advertising by restaurants that do not use trans fats. But such a suggestion ignores the difference between disseminating and absorbing information. If you have a peanut allergy, and the label on a package of cake mix says that the mix contains peanut oil, you know not to buy it; the cost of absorbing the information on the label is trivial. But if you are told that a restaurant does not use trans fats in its meals, determining the significance of that information to you would require you to undertake a substantial research project. You would have to learn about trans fats, somehow estimate the total amount of trans fats that you consume every year, estimate the amount of trans fats in the restaurant meals you consume relative to your total consumption of trans fats, and assess the significance of that consumption in relation to other risk factors that you have or don‚Äôt have for heart disease. Few people have the time for such research, or the background knowledge that would enable them to conduct it competently. Given that trans fats have close substitutes in both taste and cost, it is not unrealistic to suppose that the vast majority of people would if consulted delegate to government the decision whether to ban trans fats.
One of the great weaknesses of "libertarian paternalism" is failure to weigh adequately the significance of the operation of the cognitive and psychological quirks emphasized by libertarian paternalists on government officials. The quirks are not a function of low IQ or a poor education; they are universal, although there is a tendency for the people least afflicted by them to enter those fields, such as gambling, speculation, arbitrage, and insurance, in which the quirks have the greatest negative effect on rational decision making. As Edward Glaeser has pointed out, the cost of these quirks to officials--who are not selected for immunity to them--is lower than the cost to consumers, because the officials are making decisions for other people rather than for themselves.