Lithification happens when loose materials like dirt and sand are under high pressure for a long time. Air and liquids get squeezed out, and the material turns into a solid object. That's how rocks like limestone and sandstone are formed.
The root word, lithify, comes from the Greek word lithos, meaning "stone," plus the English suffix -ify, meaning "turn into." So lithification is the process of turning into stone. But while you could probably use it in a general sense to describe, say, what the Greek mythological character Medusa did to her victims when she looked at them, it's mostly used in geology to describe the process of how certain types of rock are formed.