Partial pooling is what a mixed effects model does to grouped data: rather than ignoring group structure or estimating every group in isolation, it shrinks each group’s estimate toward the grand mean by a factor determined by the ratio of between-group to within-group variance and the group’s sample size. Small, noisy groups get pulled hard; large groups keep their own estimates.
The idea is the regression form of the James-Stein result that shrinking a collection of estimates toward a common center beats estimating each separately, and it is equivalent to ridge regularization on group dummy variables with a penalty the data estimates itself.