Repugnant refers to something you detest so thoroughly it threatens to make you physically sick, like the idea of marrying your sister. Or wearing last year's jeans.
A repugnant thing is a thing offensive, detestable, or obscene. It can be repugnant to your mind or your morals. It can also be physically repugnant, like the smell that comes from a restaurant’s dumpster, which hasn't been emptied since the very exciting "Omelet Week," way back at the beginning of August. Makes you want to "re-PLUG-nant" your nose!