Benjamini and Hochberg developed the first (and still most widely used, I think) method for controlling the false discovery rate (FDR).
I want to start with a bunch of P values, each for a different comparison, and decide which ones are low enough to be called a "discovery", controlling the FDR to a specified value (say 10%). One assumption of the usual method is that the set of comparisons are either independent or have "Positive dependency" but I can't figure out exactly what that phrase means in the context of analyzing a set of P values.
Best Answer
From your question and in particular your comments to other answers, it seems to me that you are mainly confused about the "big picture" here: namely, what does "positive dependency" refer to in this context at all -- as opposed to what is the technical meaning of the PRDS condition. So I will talk about the big picture.
The big picture
Imagine that you are testing $N$ null hypotheses, and imagine that all of them are true. Each of the $N$ $p$-values is a random variable; repeating the experiment over and over again would yield a different $p$-value each time, so one can talk about a distribution of $p$-values (under the null). It is well-known that for any test, a distribution of $p$-values under the null must be uniform; so, in the case of multiple testing, all $N$ marginal distributions of $p$-values will be uniform.
If all the data and all $N$ tests are independent from each other, then the joint $N$-dimensional distribution of $p$-values will also be uniform. This will be true e.g. in a classic "jelly-bean" situation when a bunch of independent things are being tested:
However, it does not have to be like that. Any pair of $p$-values can in principle be correlated, either positively or negatively, or be dependent in some more complicated way. Consider testing all pairwise differences in means between four groups; this is $N=4\cdot 3/2=6$ tests. Each of the six $p$-values alone is uniformly distributed. But they are all positively correlated: if (on a given attempt) group A by chance has particularly low mean, then A-vs-B comparison might yield a low $p$-value (this would be a false positive). But in this situation it is likely that A-vs-C, as well as A-vs-D, will also yield low $p$-values. So the $p$-values are obviously non-independent and moreover they are positively correlated between each other.
This is, informally, what "positive dependency" refers to.
This seems to be a common situation in multiple testing. Another example would be testing for differences in several variables that are correlated between each other. Obtaining a significant difference in one of them increases the chances of obtaining a significant difference in another.
It is tricky to come up with a natural example where $p$-values would be "negatively dependent". @user43849 remarked in the comments above that for one-sided tests it is easy:
But I have so far been unable to come up with a natural example with point nulls.
Now, the exact mathematical formulation of "positive dependency" that guarantees the validity of Benjamini-Hochberg procedure is rather tricky. As mentioned in other answers, the main reference is Benjamini & Yekutieli 2001; they show that PRDS property ("positive regression dependency on each one from a subset") entails Benjamini-Hochberg procedure. It is a relaxed form of the PRD ("positive regression dependency") property, meaning that PRD implies PRDS and hence also entails Benjamini-Hochberg procedure.
For the definitions of PRD/PRDS see @user43849's answer (+1) and Benjamini & Yekutieli paper. The definitions are rather technical and I do not have a good intuitive understanding of them. In fact, B&Y mention several other related concepts as well: multivariate total positivity of order two (MTP2) and positive association. According to B&Y, they are related as follows (the diagram is mine):
$\hskip{10em}$
MTP2 implies PRD that implies PRDS that guarantees correctness of B-H procedure. PRD also implies PA, but PA$\ne$PRDS.