Wikipedia says:

Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups). In ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation.

In one way or two way ANOVA, (which are the types of ANOVA I have seen), the input variable $X$ is categorical, whose value represents the group of the sample.

in The Elements of Statistical Learning: Data Mining, Inference, and Prediction. by Trevor Hastie, Robert Tibshirani, and Jerome Friedman, ANOVA seems to be a way of modeling $E(Y|X)$ as sum of functions of various number of components of $X$. It doesn't relate to variance or partition of variance into some form. $X$ is not necessarily categorical either and may be continuous valued. Or am I missing something? Thanks!

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    $\begingroup$ Please give a reference for your extracted material, as a courtesy both to readers and to the original authors. $\endgroup$ – Nick Cox Jul 2 '13 at 18:57
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    $\begingroup$ I have given a link to the reference to the book ESL $\endgroup$ – Tim Jul 2 '13 at 19:00
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    $\begingroup$ At the risk of adding even more confusion, I would suggest trying to read ANOVA – Why It Is More Important Than Ever by Andrew Gelman. $\endgroup$ – Gala Jul 2 '13 at 19:17
  • $\begingroup$ Good reference. I was thinking ANOVA paper: was it Speed, was it Gelman? Answer: Both. $\endgroup$ – Nick Cox Jul 2 '13 at 19:31

One-way and two-way ANOVA are just two simple versions, but I doubt two experts on the topic would agree exactly what is central to ANOVA, treated in moderate or extreme generality.

For evidence, see Speed, T.P. 1987. What is an analysis of variance? Annals of Statistics 15: 885-910. Eleven discussions follow with a rejoinder by the author rounding it off.

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