TITLE: How does one use the Poisson summation formula?
QUESTION [55 upvotes]: While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. In fact, I've even asked people, who have probably given me perfectly good answers, but somehow their answers have never stuck in my brain. The question is simple: the Poisson summation formula is incredibly useful to many people, but why is that? When you first see it, it looks like a piece of magic, but then suddenly you start spotting that people keep saying "By Poisson summation" and expecting you to fill in the details. In that respect, it's a bit like the phrase "By compactness," but the important difference for me is that I can fill in the details of compactness arguments.
What I would like to know is this. What is the "trigger" that makes people think, "Ah, Poisson summation should be useful here"? And is there some very simple example of how it is applied, with the property that once you understand that example, you basically understand how to apply it in general? (Perhaps two or three examples are needed -- that would obviously be OK too.) And can one give a general description of the circumstances where it is useful? (Anyone familiar with the Tricki will see that I am basically asking for a Tricki article on the formula. But I don't mind something incomplete or less polished.)
For reference, here is a related (but different) question about the Poisson summation formula: Truth of the Poisson summation formula

REPLY [5 votes]: This is really a comment that could be added on to existing answers, but I can't add comments. As a particularly simple example that follows Ben's answer, as well as an example of SandeepJ's observation about changing the 'size' of the sum, consider counting solutions to some linear equation L = 0 in a subset A of a finite abelian group G. Let's say $L = x + y + z$. To count solutions to $L(x,y,z) = 0$ with $x,y,z \in A$ one could look at the sum $\sum_{(x, y, z) \in H} F(x, y, z)$, where $F(x, y, z) = 1_A(x)1_A(y)1_A(z)$ and $H \leq G^3$ is the subgroup of solutions $(x,y,z) \in G^3$ such that $L(x,y,z) = 0$. Up to some normalization factors, Poisson summation then says something like 
$$\sum_{\mathbf{x} \in H} F(\mathbf{x}) = \sum_{\chi \in H^\perp} \widehat{F}(\chi) = \sum_{\gamma \in \widehat{G}} \widehat{1_A}(\gamma)^3,$$
since $H^\perp \cong \widehat{G}$. So the 'two-dimensional' sum on the left has become a 'one-dimensional' sum on the right.
This is of course the basis of a lot of arguments in additive combinatorics, and as Ben said one might fruitfully consider the term $\widehat{F}(0)$ in the sum on the right-hand side above.