Genetic risk variants for complex, multifactorial diseases are enriched in cis-regulatory elements. Single cell epigenomic technologies create new opportunities to dissect cell type-specific mechanisms of risk variants, yet this approach has not been widely applied to disease-relevant tissues. Given the central role of pancreatic islets in type 2 diabetes (T2D) pathophysiology, we generated accessible chromatin profiles from 14.2k islet cells and identified 13 cell clusters including multiple alpha, beta and delta cell clusters which represented hormone-producing and signal-responsive cell states. We cataloged 244,236 islet cell type accessible chromatin sites and identified transcription factors (TFs) underlying both lineage- and state-specific regulation. We measured the enrichment of T2D and glycemic trait GWAS for the accessible chromatin profiles of single cells, which revealed heterogeneity in the effects of beta cell states and TFs on fasting glucose and T2D risk. We further used machine learning to predict the cell type-specific regulatory function of genetic variants, and single cell co-accessibility to link distal sites to putative cell type-specific target genes. We localized 239 fine-mapped T2D risk signals to islet accessible chromatin, and further prioritized variants at these signals with predicted regulatory function and co-accessibility with target genes. At the KCNQ1 locus, the causal T2D variant rs231361 had predicted effects on an enhancer with beta cell-specific, long-range co-accessibility to the insulin promoter, and deletion of this enhancer reduced insulin gene and protein expression in human embryonic stem cell-derived beta cells. Our findings provide a cell type- and state-resolved map of gene regulation in human islets, illuminate likely mechanisms of T2D risk at hundreds of loci, and demonstrate the power of single cell epigenomics for interpreting complex disease genetics.