The term random effects in the context of analysis of variance is used to denote factors in an ANOVA design with levels that are not deliberately arranged by the experimenter (those factors are called fixed effects), but that are sampled from a population of possible samples instead. For example, if you are interested in the effect that the quality of different schools has on academic proficiency, you could select a sample of schools to estimate the amount of variance in academic proficiency (component of variance) that is attributable to differences between schools.

A simple criterion for deciding whether an effect in an experiment is random or fixed is to determine how you would select (or arrange) the levels for the respective factor in a replication of the study. For example, if you want to replicate the study described in this example, you would choose (take a sample of) different schools from the population of schools. Thus, the factor "school" in this study would be a random factor. In contrast, if you want to compare the academic performance of boys to girls in an experiment with a fixed factor Gender, you would always arrange two groups: boys and girls. Hence, in this case the same (and in this case only) levels of the factor Gender would be chosen when you want to replicate the study.

For more information, see the description of the Analysis of Variance, Variance Components and Mixed Model ANOVA/ANCOVA, and General Linear Models (GLM) modules.