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!