How to interpret fixed/random effects in a linear mixed model (multilevel regression model)?

The fixed effect (FE) approach assumes the group effect to be correlated with the explanatory variables, where a group-specific constant term is introduced in the regression model. This constant term specific for each particular grouping in your data set embodies all the observable effects and specifies an estimable conditional mean. In simpler words, each group has its own intercept and the variability is absorbed in the disturbance term (and hence you won't get variance specific for any group). Random effect (RE) assumes the group effect heterogeneity to be uncorrelated with the included variables. RE models the group specific constant terms as randomly distributed with component of random heterogeneity specific to the j-th group and is constant across individuals within the same group e.g. students within same school is homogeneous. The reported variance components will be sigma_u denoting the estimate of the random intercept standard deviation (which belong to the within group effect), and sigma_e which is the normal residual terms. Note: I am new in this topic, but I find explanations by Greene's Econometric Analysis is very helpful in terms of understanding the theoretical parts. Rabe-Hesketh's STATA guide on Multilevel and Longitudinal Modeling is helpful in terms of interpreting the output especially if you are using STATA. Sorry I have not used R for quite a long time so can't comment on lmer.