If a creature has a "saving throws" line in the Monster Manual, use this value modifier as total bonus (it includes the stat bonus + proficiency bonus) to the die roll.
If a creature has no "saving throws" line, its bonus is based directly on the associated stat bonus (a 12 stat adds +1 to, a 16 stat adds +3 to, a 9 stat subtracts 1 from) the die roll.
A "creature" can be a Player Character or a Monster.
For Creatures = Players
Depending upon the class of the player character, to any ability score bonus or penalty for a save you would add a Proficiency bonus based on the ability score of the player character and in which ability saves the player character has proficiency. This bonus is defined in the class description for each class.
A Cleric has Proficiency for Wisdom and Charisma saves. (Proficiencies explained as helping on p. 7, Basic Rules; the Cleric save Proficiency on p. 21, Basic Rules).
Case 1. The 5th level Cleric has to roll a save versus Lightning Bolt, and he has a Dexterity of 12. He gets a +1 to save on that saving throw from his dexterity bonus. (With a Dexterity of 10, he'd get no bonus). He gets no proficiency bonus. Die roll +1 is compared to DC to see if he saves or not. (A roll of 13 would save on a spell DC of 14 or less).
Case 2. The Cleric has a Wisdom of 16 and has to roll a save
versus the Command spell. He gets +3 for his Wisdom bonus, and
another +3 for his proficiency in Wisdom saves. He adds six (+6)
to the die roll to see if he succeeds in the save or not.
Consult the PHB section for a given character class to see which saves they have proficiency in.
For Creatures = Monsters
Monsters are handled the same way (Die roll +/- stat bonus + Proficiency) but Proficiency is a bit different.
A saving throw bonus is the sum of a monster’s relevant ability modifier and its proficiency bonus, which is determined by the monster’s challenge rating (as shown in the Proficiency Bonus by Challenge Rating table). (Cite = p.4 Basic Rules(DM); MM p. 8).
Proficiency bonus are added to a monster's saving throw, with values from +2 to +9 based upon the monster/creature's CR, if the "Saving Throws" line is entered in the stat block. Otherwise, you won't use a Proficiency bonus for the save. In any case, consult the stat block for the monster for the ability score being challenged by the spell.
Example: Allosaurus (p. 9, Basic Rules(DM))
- Case 1. Save versus Lightning Bolt (Dex Save) Add the +1 Dex bonus to the die roll.
- Case 2. Save versus Shatter. (Con Save) With an Con of 17, add +3 Con to the die
Example: Green Dragon Wyrmling (p. 95 MM(DM))
Case 1. Save versus Silent Image. Int Save. +2 for Int adds +2 to
the die roll for saving versus the Silent Image conjured to distract
Case 2. Save versus Shatter. Con save. +1 Con and +2 Proficiency for
a total of +3 for a Con save (and this is reflected in the stat
(Credit to @GeneralAnders for )help with the TL/DR