Which instrument measures and records vibrations within the earth and on the earth's surface?

It is a "Seismograph". It "listens" to the vibrations, and draws out (with an inked needle) the tremors onto a piece of paper.

A seismometer is what measures and records earthquakes.

Many modern seismometers send their data directly electronically to computers, which record and automatically analyze that data instead of using the inked needle on paper.