There are many types of accelerometers, here is one type.
Force Balance Accelerometer:
It has weight that can move +/- in only one axis so in order to measure acceleration on X-Y-Z planes you will need three.
When the object being measured for acceleration starts to move, the weight will move in the opposite direction. There are two magnetic coupling devices that measure the movement and there is a summing device that will apply an electrical force in order to return the weight to the neutral point.
The strength of the electrical restoring force is measured and the electrical force is directly proportional to the accelleration. This measured electrical force is then processed and you get a readout in Gs or what ever you need to display the acceleration.
Someone else can provide the descriptions of the other types of measuring devices.