Consider a rubber band which when stretched elongates and gets back to its normal position when left as it is. This is elasticity.  Compression means change in volume when subject to pressure. Most of the liquids cannot be compressed as we do not see any volume reduction on application of pressure.

In the thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to pressure change. The property of the body to regain its original configuration (length, volume and shape) when deforming forces are removed is called elasticity.

The compressibility of a material(k)  is the reciprocal of its bulk modulus of elasticity(B). That is, k = 1/B