1.
sensitivity, tact, delicacy
The quality of being considerate and tactful about others' feelings. In Japanese, almost always used in negative contexts — to describe a lack of sensitivity.
Common patterns
デリカシー is almost exclusively used in negative constructions in Japanese. The most common pattern is デリカシーがない (to lack tact/sensitivity). It is rarely used to praise someone's sensitivity — for that, 気遣い or 配慮 would be more natural.
Etymology
From English "delicacy." The meaning has narrowed in Japanese to focus specifically on interpersonal sensitivity.