Miller was one of a handful of researchers who could be called true founders of cognitive psychology.
He is best known to undergraduates today for his 1956 paper "The magical number seven, plus or minus two," usually cited as a key paper in the understanding of short-term memory and the process of recoding (or chunking) in order to evade its limitations.
The point of that paper was actually something larger, as revealed by the subtitle: "Some limits on our capacity for processing information."
Miller commented on the short-term memory capacity limit along with other limits of about the same magnitude: our ability to distinguish simple sensory stimuli (tones or tastes, e.g.), or more complex stimuli (faces or tunes). It each case, Miller noted that our limit is about 2.5 bits, and he suggested that the similar values may be more than a coincidence. They may reflect a limitation in channel capacity, in how rapidly information may flow through the human mind.
This conception of the human as a processor of information (in the formal sense of information theory) was a radical idea, and would turn out to be enormously important as a way forward for the science, which was somewhat stymied in the 1950s by the epistemology of behaviorism.
An equally important contribution was Miller's 1960 book (with Eugene Galanter and Karl Pribram) Plans and the Structure of Behavior, which offered an explicit theory based on this new approach to psychological science. Although information theory played a small part in the work, it was an example (along with contemporaneous work by Herb Simon and Allen Newell) that rigorous research could be conducted using theories that entailed constructs (like TOTE units) that were not directly observable.
In recent years Miller's work focused on the psycholinguistics, especially word meaning (summarized in this wonderful review) and the organization of meaning in long-term memory, modeled in a lexical database, WordNet.
Miller's influence may be understood by the fact that he was awarded more or less all of the prizes and honors a psychologist might win, culminating in the National Medal of Science in 1991.