Each person experiences the world through a unique conceptual lens, shaped by personal experiences, natural variations, or disease. These individual differences have remained largely inaccessible to cognitive neuroscience and clinical neurology, limiting the development of precision medicine approaches to cognitive disorders. To overcome this limitation, here we develop a new statistical framework to measure and interpret individual differences in functional brain representations. We apply this framework to characterize how different individuals represent the same concepts. Twenty-four participants listened to narrative stories while their brain activity was measured with functional MRI (fMRI). Encoding models were used to recover how hundreds of concepts were represented in each person's brain. Despite listening to identical stories, participants showed systematic individual differences in conceptual representations. These differences reveal person-specific biases in how concepts are represented in the brain. Individual variability was highest in regions that represent social information. Because these regions are thought to integrate sensory information with personal beliefs and experiences, the observed individual differences may reflect cognitive traits unique to each person. Our work reveals that individual differences are a systematic, measurable principle of conceptual representations in the human brain. By enabling researchers to measure and interpret differences in person-specific functional brain representations, our work establishes a new paradigm for precision neuroscience. This paradigm provides a rigorous foundation for developing fMRI applications in precision medicine to diagnose and monitor cognitive disorders.