Deformation occurs at any point of the curve, not just at A.
Point A marks the beginning of permanent set or plasticity. It is called the Elastic Limit or the Limit of Proportionality.
Point B marks the Yield Point.
Beyond Y the stress decreases with increasing strain and necking occurs. The stress at Y is the ultimate tensile strength.
X is the final fracture point.
The slope of the curve is the modulus of elasticity. (Youngs Modulus).
If the material is stressed beyon the elastic limit at A to point B and then destressed, the material will be work hardened (strain hardened) so that B is now the elstic limit and the new loading curve will follow the right hand dashed line. It will also have acquired a permanent set equal to the difference between the two dashed lines along the strain axis.