A gravimeter is an instrument used to measure gravitational acceleration. Every mass has an associated gravitational potential. The gradient of this potential is an acceleration. A gravimeter measures this gravitational acceleration.
The first gravimeters were vertical accelerometeres specialized for measuring the constant downward acceleration on gravity on the earth's surface. The earth's vertical gravity varies from place to place over the surface of the Earth by about +/- 0.5%. It varies by about +/- 1000 nm/s^2 ("nanometers per second squared") at any location because of the changing positions of the sun and moon relative to the earth.
The change from calling a device an "accelerometer" to calling it a "gravimeter" occurs at approximately the point where it has to make corrections for earth tides.
Though similar in design to other accelerometers, gravimeters are typically designed to be much more sensitive. Their first uses were to measure the changes in gravity from the varying densities and distribution of masses inside the earth, from temporal "tidal " variations in the shape and distribution of mass in the oceans, atmosphere and earth.