What are the key statistical concepts that relate to data mining? What topics in statistics are most useful/relevant to data mining?
 A: Understanding multivariate normal distribution http://en.wikipedia.org/wiki/Multivariate_normal_distribution is important. 
The concept of correlation and more generally (non linear) dependence is important. 
Concentration of measure, asymptotic normality, convergence of random variables.... how to make something from random to deterministic! http://en.wikipedia.org/wiki/Convergence_of_random_variables
maximum likelihood estimation http://en.wikipedia.org/wiki/Maximum_likelihood and before that, statistical modeling :) and more generally minimum contrast estimation. 
stationary process http://en.wikipedia.org/wiki/Stationary_process and more generally stationnarity assumption and ergodic property. 
as Peter said, the question is so broad ... that the answer couldn't be given in a post ... 
A: Understanding the causes of over-fitting, and how it can be avoided, is absoluetly  vital in datamining.  If you mine for statistical associations in data, you will always be able to find them (even if the data are purely random) but that doesn't mean they are of any predictive value.  The more you mine, the more likely you are to dig up a spurious rule.  I know that may sound rather obvious, but that doesn't mean that spurious rules never get presented to clients! ;o)
