Having finesse means you can handle difficult situations with diplomacy and tact, like the finesse it takes to help two friends work out their differences — without taking sides or alienating either one.
Finesse is having grace under pressure. It's handling the rantings of an angry customer with a smile and a calm tone. Someone who has finesse says the right thing at the right time — or knows when to say nothing at all. Finesse looks like fineness and in fact comes from the Middle French word that means exactly that — delicate in nature. People with finesse can handle anything — with a delicate approach that really works.