What instrument do scientists use to measure earthquakes and other movements of the earth?

A seismograph.

A longer answer

A seismograph or seismometer is one kind of instrument that is used to measure the effects of earthquakes. There are a number of different types of seismometers and they all work in slightly differing ways but generally are based around the same principle.

That principle is inertia. Inertia basically means that stationary masses will remain stationary until a force is applied to them. Conceptually a seismometer can be thought of as a weight or mass, suspended by springs in a frame which is bolted to the floor.

When an earthquake occurs the frame which is attached directly to the floor is caused to move by the seismic waves, however the mass / weight on the spring does not as the seismic waves do not directly affect it (their effect is dampened by the inertia of the mass and the springs). If you were to attach a pencil or pen to the weight and a piece of graph paper to the frame so that they were touching the movement of the frame relative to the stationary mass would be recorded.

Modern seismometers are much more complex than this but in essence rely on this principle.

During an earthquake there can also be substantial permanent ground deformations. As such, seismologists will attempt to measure the movement of the crust around a fault zone to estimate the total strain that has accumulated. This measurement is often made using high precision GPS to measure relative positions of surface features around the fault zone. They may also use terrestrial laser scanning equipment which can measure changes in the ground shape (recording deformation) or by using a special form of radar and a technique called synthetic aperture radar interferometry (InSAR for short). This process essentially involves the use of a radar to create a series of very accurate relief maps of the ground surface over time and then to compare the maps to create a final plot showing the changes between them which is a record of the land surface deformation.

They may also use strain gauges and tilt metres within boreholes to observe ground deformations as well as a technology known as time domain reflectometry which is an electrical technique used to locate damage and deformation in electrical cables and which in turn can be used to measure subsurface deformations.