What is the difference between "risk" and "uncertainty"? In what situation should I use each word?
Risk and Uncertainty have a specialized meaning in the financial world.
The Wikipedia article on Risk has an extended quote from Risk, Uncertainty, and Profit by Frank Knight (1921). He implies that risk is sometimes mathematically quantifiable, but uncertainty is never quantifiable.
By Knight's reasoning, you would use risk when examining a stock's previous return to predict its future return. If a stock were volatile in the past, one might expect that it would be more volatile in the future and thus have a higher risk. If the price changes of a stock were small and predictable, then one might expect that the price changes would continue to be less volatile in the future and thus have a lower risk.
There is a mathematical concept called expected value that can be used to quantify the risk. It may be based on a probability (such as a coin turning up heads 50% of the time). It may instead be based on statistics (using a concept of variance, which is related to standard deviation). Financial risk is based on expected value, and is quantifiable.
In contrast, by Knight's reasoning, uncertainty would not be quantifiable. The returns of both the volatile and the steady stock in the future are neither knowable nor quantifiable, so the returns are uncertain.
The face that turns up on a die before it is cast is unknowable and therefore uncertain. The probability that it will turn up a 1, 2, or 3 is quantifiable (it will happen in half the cases). So there is a known risk in casting the die and winning $10 if it turns up 1, 2, or 3 while losing $10 if it turns up 4, 5, or 6.
The difference can be described as follows:
Risk is concerned with the outcome of events, not necessarily dangerous. Uncertainty is concerned with its probability.
That is, if you're more concerned about the outcome associated with some event, you estimate its risk, but if you're interested in the probability of an event, you estimate its uncertainty, where uncertainty is the opposite of likelihood.
In risk assessment courses for industrial use, you learn to estimate risk and uncertainty independently. The risk from a fire may include loss of life, damage to property, interruption to production. The uncertainty is a numerical value which estimates its likelihood.
If you do something that is dangerous, there´s a risk you might get hurt.
If you do something that is safe but difficult to do, the outcome may be uncertain. (but it is not risky)
Try to learn to make computer programs in c++ programming language. Outcome is uncertain. Try to learn how to tame lions. That is risky!
It is uncertain if there is enough ice on planet Mars to make enough water and oxygen for a manned mission. It may be too risky for people go to planet Mars.
Hope that clears it up a bit.
EDIT: I stand corrected by the other answer. I went ahead to answer the question even though there was a bit of a risk my answer was going to be inaccurate. Yet of course there was no actual danger.
Risk is when we do not know what the outcome is, but we do know the distribution but uncertainty the outcomes, whereas uncertainty is when we do not know what the outcome, and we do not know the distribution.
The practical difference between the two categories, risk and uncertainty, is that in the former the distribution of the outcome in a group of instances is known (either through calculation a priori or from statistics of past experience), while in the case of uncertainty this is not true, the reason being in general that it is impossible to form a group of instances, because the situation dealt with is in a high degree unique.
Risk, Uncertainty and Profit, Frank H Knight, pub. Houghton Mifflin 1921