ANOVA does not assume that the entire response column follows a normal distribution. ANOVA assumes that the residuals from the ANOVA model follow a normal distribution. Because ANOVA assumes the residuals follow a normal distribution, residual analysis typically accompanies an ANOVA analysis. Plot the residuals, and use other diagnostic statistics, to determine whether the assumptions of ANOVA are met.
You can evaluate the assumption that the residuals follow a normal distribution from the response data when the data do not include a covariate. In ANOVA, the entire response column is typically nonnormal because the different groups in the data have different means. If the data for each individual group follow a normal distribution, then the data meet the assumption that the errors follow a normal distribution. This condition is typically stronger than the condition that the residuals follow a normal distribution. If the groups contain enough data, you can use normal probability plots and tests for normality on each group.
The ANOVA procedure with fixed factors and equal sample sizes works quite well even when the assumption of normality is violated, unless one or more of the distributions are highly skewed or the variances are very different.