For example a Cleric casts Inflict Light Wounds on Target ‘A’ whom can make a saving throw for half damage. Does Target ‘A’ make this saving throw immediately or at the start of their turn? A Wizard casts Hold Person on Target ‘A’ (unlucky fellow) does they get to make an immediate saving throw and a saving throw at the start of their turn?
The saving throw is made when the spell takes effect.
For cure light wounds the saving throw happens when a successful touch attack is made.
For hold person, the save happens when the spell is cast. Hold person also allows additional saves a the beginning of the subjects turn.
The only exception I can think of are illusion spells. Illusion spells allows a save when you interact with them.