Saving throws are an 'equal or exceed' roll, so a DC of 15 could be beaten with a d20 roll of 15-20.
An attempt to resist a spell, trap, poison, disease or similar.
Forced upon you by the situation.
d20 rolled with the stated ability modifier.
Each class has proficiency in at least two saving throws, i.e. they add their Proficiency Bonus.
Difficulty Class of the throw is determined by the effect that caused it, e.g. a spell's DC saving throw is determined by the caster's own spell save DC. In the case of a Cleric, this is 8 + their proficiency bonus + their Wisdom Modifier.
A success or failure's effects are usually detailed by the initiating effect, in the above example of a spell, it might be that successfully making the save results in half damage being taken, rather than the full amount.