To inscribe means to write something in a permanent or formal way. When you have a jeweler inscribe your initials inside your new ring, she uses a machine to carve them into the metal.
A stone carver might inscribe your grandfather's name on a memorial stone, and your best friend might inscribe his name and a favorite quote in the back of your yearbook. In geometry, to inscribe is to fit one shape inside another, like when a triangle fits neatly inside a circle with its points just touching the perimeter. The Latin root of inscribe is inscribere, "to write in or on."