How do you interpret a significant but weak correlation? I have carried out a number of Pearson Chi Square tests on the relationship between tail lesions and abscessation in pigs. I have found a significant (p = 0.005) but weak (r = 0.288) association. How do I interpret this?
For example, does this mean that there is a clear relationship between the two but that it is just on a small scale?
 A: You could say it like this:

The association is small, but not zero. 

However, I don't know that I'd call a V of 0.288 "small". 
Don't confuse "statistically significant" with important. Statistical significance is very different from practical importance.
A: Adding on to @Glen_b 's excellent answer and your comments there.
As he said, it is significant but weak because the sample size is large enough to make a small effect significant. But now you need a way of showing the effect size. V is one such way, but it isn't intuitive to many and is not as well known as some other measures.
From your question, it seems like you have a 2x2 table of lesions (yes/no) and abscess (yes/no). You could then show that table; you could give the odds ratio or any of a number of other measures for 2x2 tables. 
A: More meaningful in this case is the $\text{R}^2$ which explains the proportion of variation in your observations accounted by the association. For example if your $R$ was 0.1 (p= 0.005) due to the large sample size, it means 1% of the variation in tail lesions in pigs is accounted for by abscesses. In a multifactorial situation such associations though informative may not be very meaningful. Again be cautious since correlation does not imply causation.
A: I like to comment on the perceived usefulness of the relationship.  For instance:

The association is statistically significant but not practically relevant.

That being said, your association doesn't seem to be that weak.
