The Malmquist bias is a selection bias applicable to surveys stemming from missing the dimmest objects. The longer the distance, the larger percentage of objects at that distance will be too dim to be detected, and the collected observations will include both bright and dim objects at nearer distances but only the brighter ones at greater distances. In other words, the greater the distance, the stronger the preference (bias) toward brighter objects, meaning evaluation of the randomness of sample-sets must take this into consideration. Generally the brightness is associated with the type or size of the object, and population statistics at different distances need to take the bias into account.
This is in contrast to another type of observation bias, the Eddington bias. I've seen references that call two terms synonymous, but to my understanding, the Eddington bias stems from brightness measurement errors (or perhaps "errors" from seeing, reddening, extinction and the like), whereas the Malmquist bias stems from distance alone and applies even if there are no such brightness mistakes. However, the means of compensating for the two biases might be equivalent.