The nested error regression model is a useful tool for analyzing clustered (grouped) data, and is especially used in small area estimation. The classical nested error regression model assumes normality of random effects and error terms, and homoscedastic variances. However, these assumptions are often violated in real applications and more flexible models are required. This article proposes a nested error regression model with heteroscedastic variances, where the normality for the underlying distributions is not assumed. We propose the structure of heteroscedastic variances by using some specified variance functions and some covariates with unknown parameters. Under the setting, we construct the moment-type estimators of model parameters and some asymptotic properties including asymptotic biases and variances are derived. For predicting linear quantities including random effects, we suggest the empirical best linear unbiased predictors and the second-order unbiased estimators of mean squared errors are derived in the closed form. We investigate the proposed method with simulation and empirical studies.