Think of a weight hanging on a spring or piece of elastic (known as a spring-mass system). If you prod it, it will start to bounce up and down, oscillate, vibrate. The reason it does this is that there is energy (from your finger) stored in the system, and the energy is alternately transferred between kinetic energy as the mass moves, and potential energy as it stretches the spring. If that were all there is to it, the oscillation would continue forever, but there is also energy dissipated both by wind resistance, and frictional losses in the materials the system is made from. This loss of energy causess the amplitude of the oscillations to gradually decay away until the oscillation stops altogether. This process is known as damping. The more energy is lost with each cycle of oscillation, the faster the oscillations will stop, so with a fairly low viscosity fluid like air the oscillation dies slowly, but higher viscosity water will damp the system more, so the oscillation will die away faster.
A bell is exactly the same system in principle, except that the spring and mass are both integrated in the same component, and the frequency of oscillation is higher, so you hear it rather than see it. This is an example of what oscillations look like as you progressively increase the damping factor: