Risk has been studied from a variety of perspectives, which can most easily be categorised as the psychological approaches to risk and the social scientific approaches to risk [link - to come]. Both traditions are concerned with finding out more about how people percieve and understand risk, and how they react towards it, though they differ in their interpretations of what risk actually is, and in their ideas of what if anything should be done about their insights to risk.
The meaning of risk
The word "risk" itself means a combination of the probability of something happening combined with the expected benefits or harms of it. Therefore a risk is not merely a danger or a probability itself, but it also takes into account other factors surrounding uncertainty. What exactly these other factors are, often depends on who does the analysing and in what circumstances. Very often the benefits are measured in expected gain (or loss) of money or other commodities, and risk is therefore quantifiable. This is often being done for example when we analyse risks in gambling. Note also that risk, at least in this original sense, can mean both negative and positive outcomes, though the positive connotation of risk has largely disappeared in everyday language.
Measuring expected gains and losses however is difficult when there are other things than money at play. First of all there is can be an emotional benefit of, say gambling, even if the risks are (monetarily speaking) high: they could be balanced out by the pleasure we get from the gambling experience, and therefore gambling, even if in the long run it leads to monetary losses, is not neccessarily an irrational activity. The problem for quantifying these deeper meanings of risk is that we would need to compare two different kinds of things, and money and "emotional value" are in this respect like apples and oranges.
The problem is much less abstract for the insurance industry and more general risk management, which needs to put a monetary value on non-monetary things. When it comes to health risks, how do we even start putting a monetary value on human lives?
Away from the more technical meanings of risk, the word "risk" has taken on a different meaning in everyday conversation, by being often used to mean "danger" or "hazard", with only a vary vague connection to quantifiable uncertainty, or probability. For example there may be talk about the risks of GM food: In this case there is no attempt to put a numerical answer to the danger. This may well be an impossible task. To take Donald Rumsfeld's memorable phrase, the idea of "GM risks" includes not only known unknowns (the dangers that are being debated at the moment), but also the "unknown unknowns" (such as unintended consequences of the new technology that we have no idea about yet).
Research on people's reaction to risk
One early strand of research on people's reactions to risk involved questioning how people perceive risk and how they respond to it. This line of study (outlined further here), has been conducted mainly by psychologists and has led to a number of thriving areas of research covering for example risk management for insurance (i.e. why do people insure for some risks, and not others) or clinical risk communication (e.g. how should doctors best communicate the risks of certain drugs or procedures to patients).
It is often argued that at the heart of this tradition of research there lies the assumption that risks are real and calculable, and that people who don't recognise them properly suffer from cognitive defects that prevent them from recognising the real risks they are under. The experts on the other hand are trained in the correct statistical methods and are therefore able to judge risks correctly, and that therefore people should just follow what the experts tell them.
Instead, social scientists have pointed out (link! to sociology of risk to come) that talk about risks often performs other rhetorical duties such as demarcating group boundaries and ostracising at-risk communities by effectively blaming them for not managing their risks effectively. Other sociologists have noted that concern about risks is a peculiarly late modern phenomenon that reflects how contemporary society thinks about itself and the world, and that effectively dealing with risks and risk perception must take this into account.
However, the usual way that social science risk research portrays the psychology of risk tradition is slightly unfair: First of all, psychology of risk research may have started out with the above assumptions, but most have by now accepted the criticism from social scientists that experts do not always know better and that risks are socially situated as well as not always clearly definable at all (see here [link]). Quite early on, for example, studies have shown that experts are prone to exactly the same sorts of errors of estimating risks and probabilities as laypeople (see for example some of the essays in the Kahnemann & Tversky collection, ref to come).
A fundamental difference between most psychological and most social scientific conceptions of risk is to do with a more philosophical attitude towards the reality of what risks actually are. Lupton
The commonly held caricature is that most psychological research tends towards the realism end of this characterisation, while the social sciences tend more towards the social constructivism end. In reality, while both sides often fling accusations at each other that the other is situated at an extreme end of this spectrum, most researchers in fact favour a weak social constructivism of some sort, though here I am commenting my own observation rather than summarising the literature: At the end I believe that whatever philosophical differences there are between the approaches outlined here, they talk about essentially different things which can easily be combined for a fuller understanding of what risk means and how it is used.