How do you find causal relationships in data? Lets say I have a table with columns "A", "B"
Is there a statistical method to determine if "A" causes "B" to happen? One can't really use  Pearson's r, because:


*

*it only tests the correlation between values

*correlation is not causation 

*Pearson's r can only correlate linear relationships


So what other options do I have here?
 A: The answers and comments so far are basically correct at the practical level, but for completeness, there is research into so-called causality models that are based upon Bayesian statistics and graph theory.  So although in general correlation indeed does not imply causation, there are more complex models that do attempt to tease out causation.  See the book Causality by Judea Pearl for more details, but this is very heavy-duty mathematics and is probably not what you want.
A: There are many so-called quasi-experimental methods with which you can credibly argue about causality, even though your data are observational. These methods typically rely on finding a source of exogenous variation in your variable of interest. 
I think a good and accessable overview is given in the book "Mostly Harmless Econometrics". They cover basically all quasi-experimental methods that people (meaning: economists) believe in (at least sometimes). They do not cover the methods mentioned by for instance trb456 (for the same reason: not many believe in them).
A: To determine causation you need to perform a randomization test. You take your test subjects, and randomly choose half of them to have quality A and half to not have it. You then see if there is a statistically significant difference in quality B between the two groups.
It is important that you do the randomization before you do any measurement. In particular, if you are given a data set with $A$ and $B$ already measured, then it is impossible to determine causation.
Note that it may be impossible to do the randomization test that you want to do. For example, how could you test if being tall causes you to weigh more? Certainly there is a correlation between height and weight, but you can't randomly assign one group of people to a 'tall' group and one to a 'short' group. In this case, the randomization test can't be done.
A: Somers' d works for explaining the relationship between ordinal variables in a way that pearson's correlation coefficient does for data sets.
