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I need to explain the intuition behind what ANOVA is doing to a non-technical person. Is there a visual that explains the idea? A visual that illustrates the key idea in the context of a one-way ANOVA with perhaps 3 factor levels might be helpful?

Let us suppose that the person has taken some statistical courses as a student in the distant past but has forgotten the details of even performing a z-test. However, he/she remembers that hypothesis testing is used to check if the observed effects are due to random chance or due to a real change in the parameter of interest.

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Please add to your question what you expect the listener to have for statistics knowledge. Do they know a t-test? Do they have some familiarity with hypothesis testing?... etc. If they have absolutely none there's not much detail I'd ever want to go into. – John Oct 16 '12 at 20:35
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Of possible interest: How to visualize what ANOVA does? – chl♦ Oct 16 '12 at 21:19
How about using an example of three firework exploding in the sky? The three explosion heights would be their group means. Given the same three heights, the difference can feel greater if the firework explosion radius is small (low within group variation); and they can also feel indifferent if the explosion radius is huge relative to their explosion height differences. Hence, both explosion heights (between) and explosion radius (within) have to be considered. – Penguin_Knight Oct 17 '12 at 0:18

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ANOVA is statistical technique used to determine whether a particular classification of the data is useful in understanding the variation of an outcome. Think about dividing people into buckets or classes based on some criteria, like suburban and urban residence. The total variation in the dependent variable (the outcome you care about, like responsiveness to an advertising campaign) can be decomposed into the variation between classes and the variation within classes. When the within-class variation is small relative to the between-class variation, your classification scheme is in some sense meaningful or useful for understanding the world. Members of each cluster behave similarly to one another, but people from different clusters behave distinctively. This decomposition is used to create a formal F test of this hypothesis.

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