Best linear unbiased prediction (BLUP) is a way of predicting something (like the weight of a animal, for example) by using a combination of the average weight of all animals of the same type, and the differences in weight between similar animals.
Think of it like this: You have a lot of animals of the same type, and you want to know what weight an individual animal would be. You look at the average weight of all the animals and then you look at the differences between the weights of similar animals. You then combine these 2 things to make a prediction of the individual weight of the animal. That's BLUP!