A Pocket Square & Handkerchief
The pocket square is for show and belongs in your jacket breast pocket. It is normally made from silk, a light-weight cotton or linen, and should be small enough to fold without creating bulk.
The handkerchief is for blow and goes in your back or front pants pocket or inside your jacket lower pockets/inside pockets. It is normally made from cotton so it can be washed and used to absorb fluids and can be larger in diameter as bulk is less of a factor.