Robust standard errors account for heteroskedasticity in a model’s unexplained variation. That is, if the amount of variation in the outcome variable is correlated with the explanatory variables, robust standard errors can take this correlation into account. Robust standard errors are useful in social sciences where the structure of variation is unknown, but usually shunned in physical sciences where the amount of variation is the same for each observation. Robust standard errors are generally larger than non-robust standard errors, but are sometimes smaller.
Clustered standard errors are a special kind of robust standard errors that account for heteroskedasticity across “clusters” of observations (such as states, schools, or individuals). The clustering is performed using the variable specified as the model’s fixed effects. Clustered standard errors are generally recommended when analyzing panel data, where each unit is observed across time.