Saving throws
For a comprehensive summary of the mechanics behind all rolls and modifiers, see: Die Rolls
A Saving Throw, also called a save, is a D20 roll the game makes for a creature to decide whether it can save itself against a special kind of threat, such as a spell, poison, trap, and so on. Each save has a Difficulty Class (DC) which must be reached or exceeded for the save to be successful.
The Difficulty Class
If a creature casts a spell that forces another to make a save, the DC is determined according to the properties of the caster, using the following formula:
Difficulty Class = 8 + Ability Score Modifier + Proficiency Bonus
The Ability Score used to determine the Modifier depends on the class of the caster; see Attack Roll. The Proficiency needed to gain the Proficiency Bonus is spell casting. All creatures capable of casting spells through their own abilities have this proficiency, but if a creature that can't normally cast spells uses a Scroll to cast one, it will lack the Proficiency Bonus. (TODO: Verify that this is actually implemented in BG3!)
In case of threats that don't originate from a spell caster, such as a trap that just sprung or a poisonous apple you just ate, the individual threat in question determines the DC. For example, a rather ineffective trap could have a DC of just 5, whereas a very powerful one could have a DC of 15. A slightly spoiled tart could impose a Constitution Save of DC 5, whereas a potent venom from a snake could impose a DC 15 Constitution Save on the hapless victim.
Modifiers on the Saving Throw
Each Saving Throw is related to a specific Ability Score. (Depending on the ability, the Saving Throw can be referred to as a Strength Save, Dexterity Save, Constitution Save, and so on.) The type of save determines the Modifiers the creature uses. In addition to its Ability Score Modifier, the creature can also add its Proficiency Bonus to the result if it has Proficiency in the corresponding type of Saving Throw. This means the formula is:
Result =d20 + Ability Score Modifier + (optional) Proficiency Bonus
Each Class provides a creature Proficiency in two types of Saving Throws.
The save succeeded, but I still got hurt!
Making a successful Saving Throw could mean avoiding negative effects entirely, or it could merely mean halving the damage taken. Depending on the threat, there could also be other rules that determine what happens on a failed or successful save.