Three ways Hume was really influential in psychology were:

1. Hume said simple ideas could be built up into complex ideas through some simple laws of association, instead of depending on the mind playing an active role with its unexplained homunculus abilities. PDP models of information processing are based on applying a few generic rules to a large collection of simple processing units instead of depending on a program that executes an explicit set of instructions.
2. Hume said all knowledge could be reduced to either matters of fact or observation (synthetic statements) or matters of the relations among ideas (analytic statements), and anything else didn't count as knowledge at all. Logical positivism was the philosophy of science in the 20th century that took that as its starting point and said science was the model for all knowledge; it was the most influential philosophy underlying behaviorism, affecting the kinds of theories that were developed and favoring the neglect of any role for the mind.
3. Hume said that cause and effect couldn't be perceived with the senses and were merely the result of observing repeated contiguity between successive events, along with the unjustifiable assumption that the future will be like the past; therefore the best attitude toward science would be to stick to cataloging the observations and their typical sequences without proposing any underlying causal explanations of why things happen. The most influential behaviorist, B.F. Skinner, adopted much the same attitude by arguing against complicated theorizing in terms of physiology or cognition or other unobservable variables, and instead just cataloging the observed relationships between stimuli, responses, and reinforcements and punishments.