A handkerchief is a square of fabric that you can use to blow your nose. You can also tuck a handkerchief in the breast pocket of a suit, or wear it tied jauntily around your neck.
You can call a handkerchief a "hanky," or just a kerchief, although this more often means "a square piece of cloth worn as a head covering." This last part of handkerchief comes from the Old French couvrechief, literally meaning "cover head," from couvrir, "cover," and chief, "head." So a handkerchief is a similar cloth held in the hand — to mop your brow or wipe your eyes.