Sensibility is your ability to feel or be aware of something. Your artistic sensibility might allow you to understand even the most complex work of art.
Sensibility often refers to your ability to respond to emotions or to appreciate artistic works — things that you sense rather than analyze logically. Your poetic sensibility might allow you to choose exactly the right word to create a particular tone when you’re writing. Sensibility can also refer to your sensitivity to good or bad impressions — it’s sort of like your feelings. Used in this way, the word is often plural. Your brother’s thoughtless remarks may offend your sensibilities.