Sensitivity has many shades of meaning but most relate to your response to your environment — either physical or emotional. Sensitivity to cold means you’re quick to notice the slightest chill — in the air or in another person’s actions.
A sensitivity to pollen means you’re sneezing any time it’s in the air — you just always seem to react to it. It’s the same with emotions — sensitivity means you pick up on the feelings of others. If you have great sensitivity to your classmates, then you’re aware of their needs and behave in a way that makes them feel good. If you introduce a new policy with sensitivity, that means you consider how others will react to it.