Viscosity:
Viscosity is the property of a fluid that determines its resistance to shearing stresses. It is a measure of a fluid's internal friction or the "stickiness" of the fluid. When a fluid is subjected to a shearing force, such as when it flows or is deformed, the layers of the fluid slide past each other. The resistance to this sliding motion is directly related to the fluid's viscosity.

Surface Tension:
Surface tension is the property of a liquid that allows it to resist external forces and minimize its surface area. It is caused by the cohesive forces between the liquid molecules at the surface. While surface tension plays a role in the behavior of fluids, it is not directly related to their resistance to shearing stresses.

Compressibility:
Compressibility is the measure of how much a fluid can be compressed or how much its volume can be reduced under pressure. It is related to the density of the fluid and its ability to change volume when subjected to external forces. Compressibility is not directly related to a fluid's resistance to shearing stresses.

Explanation:
The correct answer is option 'A' - Viscosity. Viscosity is specifically defined as the resistance of a fluid to flow or deformation under an applied force or stress. It is a measure of the internal friction in a fluid. When a fluid is subjected to a shearing stress, the layers of the fluid slide past each other. The resistance to this sliding motion is directly related to the fluid's viscosity.

Viscosity is influenced by factors such as temperature and the nature of the fluid itself. For example, honey has a higher viscosity than water. Fluids with high viscosity flow more slowly and are more resistant to deformation, while fluids with low viscosity flow more easily and are less resistant to deformation.

Measuring viscosity is important in various industries and applications. For example, in the oil industry, viscosity is a crucial factor in determining the flow characteristics of crude oil and its ability to be transported through pipelines. In the manufacturing industry, viscosity is important in the formulation and production of paints, adhesives, and other materials.

In conclusion, viscosity is the property of a fluid that determines its resistance to shearing stresses. It is a measure of a fluid's internal friction and is directly related to its ability to flow or deform under an applied force. Surface tension and compressibility are separate properties of fluids that are not directly related to their resistance to shearing stresses.