The BLUP is what a mixed model reports for each group’s random effect. It is called a prediction rather than an estimate because a random effect is an unobserved draw, not a parameter; the BLUP is its conditional expectation given the data, which for a random intercept works out to the group’s raw deviation multiplied by a shrinkage factor of tau squared over tau squared plus sigma squared over the group size.
The formula encodes calibrated trust: groups with lots of data keep their own estimates, tiny groups are pulled most of the way to the grand mean, and the crossover is governed by the ratio of between-group to within-group variance. BLUPs originate in animal breeding, where predicting the genetic merit of individual animals from sparse records is exactly this problem.