Something that's dying is about to die or is getting ready to stop existing. A shriveled up plant that hasn't been watered in days is probably dying.
Dying is a process that will happen to us all: living things eventually die. But non-living things can be dying, too: a battery that doesn't have much juice left is dying, and so is a TV show that's bombing in the ratings. A craft that's losing popularity can be called a dying art. States of extreme stress or anticipation can also be described as a form of dying, as in "I'm dying to know how this book ends!"