A

Step 1: Understand the definition of surface tension.
Surface tension is defined as the force that acts on the surface of a liquid, causing it to behave like an elastic sheet. It is quantitatively described as the work done per unit area to increase the surface area of a liquid.

Step 2: Analyze the conditions under which surface tension is measured.
Surface tension is typically measured under isothermal conditions, where the temperature remains constant. In these conditions, the energy input required to increase the surface area is consistent.

Step 3: Review the options provided:
- Option A: The work done per unit area in increasing the surface area of a liquid under isothermal condition - This is the correct definition.
- Option B: The work done per unit area in increasing the surface area of a liquid under adiabatic condition - This is incorrect because adiabatic processes involve changes in temperature and do not define surface tension accurately.
- Option C: The work done per unit area in increasing the surface area of a liquid under both isothermal and adiabatic conditions - This is also incorrect, as surface tension is specifically associated with isothermal conditions.
- Option D: Free surface energy per unit volume - This is not a correct definition of surface tension, as it focuses on energy rather than work done to increase surface area.

Conclusion: Therefore, the correct answer is Option A.