Generally it is used in the context of someone’s social awareness, and their ability to sensitively manage tricky social situations.
Someone with Fingerspitzengefühl is diplomatic and tactful, and knows what to say at the right time, for example, a manager who successfully arbitrates between feuding colleagues, or a friend who gives you space because they understand you don’t want to speak right now.
Someone who is thoughtless and impudent is normally lacking Fingerspitzengefühl. This could be your boss who loudly criticizes your work in front of your colleagues, or your boyfriend who tells you that you do look fat in that dress.
Fingerspitzengefühl can also be used in a more literal sense, to mean the care and dexterity required to carry out nimble and delicate work with your hands, such as that of a watchmaker or potter.
Nadelarbeit verlangt großartiges Fingerspitzengefühl.
Needlework demands a great dexterity.
Du musst an der Situation mit Fingerspitzengefühl herangehen.
You have to approach the situation with tact.