The classic example of this is "When ice cream sales are high, so are murder rates. So ice cream consumption must cause murders." Of course, there's no causation there; ice cream sales and murder rates just both happen to go up when the weather is hot. Other examples can be found here:
http://www.don-lindsay-archive.org/skeptic/arguments.html#correlationThere are a lot of arguments about the dangers of obesity that rely on this fallacy. People assume that because there's some (but not as much as we're lead to believe by the popular media) correlation between increasing weight and increasing incidents of certain diseases and premature death that obesity causes these diseases and premature death. But when you factor out certain lifestyle elements, such as activity, consumption of fruit and vegetables, and smoking, the picture becomes more complicated. It's more likely that lack of activity and poor eating habits both contribute to obesity and to ill health, not that obesity causes ill health in and of itself, since people who are obese but who exercise regularly have virtually the same life expectancy as thinner, active people.
This fallacy has also mislead some people to be anti-vaccination because of the supposed correlation between exposure to mercury in vaccines and the rising rates of autism. As more children have been vaccinated, the argument goes, more cases of autism have occurred. Therefore, the mercury in vaccines must be causing autism. There are no well done studies that show any link between the mercury in vaccines and autism. What is more likely going on is that over the years, screening is catching more cases of autism in children AND the criteria for diagnosing mild forms of autism have broadened, resulting in larger numbers of children being
diagnosed as autistic, not necessarily that there are many, many more causes of autism occurring than in previous years.
On the other hand, you can't entirely ignore strong correlations in medical studies; this is exactly how we found out that tobacco smoking causes cancer. The strong correlative evidence between cigarette smoking an rates of cancer is something like *
900 times stronger thank the link between obesity and most diseases it supposedly contributes to. And, we also understand something more about the mechanism of how tobacco smoking causes cancer, which lends additional evidence besides the correlative evidence to the smoking/cancer link.
* I remember the numbers being something close to that according to Glenn Gaesser in "Big Fat Lies" but I don't have the actual number in front of me. Post to be updated with exact number when I can track it down.