If a character attempts something where the outcome is uncertain and failure has consequences, they make a saving throw, or “save”. To make a save, add the bonus of the relevant ability to a d20 roll. If the total is greater than 15, the character succeeds. If not, they fail.
Designer's Note: Requiring saves to exceed 15 means that new PCs have around a 25% chance of success, while level 10 characters have around a 75% chance of success, since ability bonuses can get up to +10 by level 10. This reflects the general pattern found in the save mechanics of early D&D.
If the save is opposed by another character, then instead of aiming to exceed 15, the side doing the rolling must get a total greater than the opposing character’s relevant defense score in order to succeed. If they fail, the opposing side succeeds. This type of save is called an opposed save. Note that it doesn’t matter which side does the rolling, since the odds of success remain the same.
Example: A wizard casts a fireball spell at a goblin, who gets a saving throw to avoid. This is resolved as an opposed save using the wizard’s Intelligence versus the goblin’s Dexterity. The goblin may roll plus their Dexterity bonus, hoping to exceed the wizard’s Intelligence defense or the wizard may roll plus their Intelligence bonus, hoping to exceed the goblin’s Dexterity defense.
Designer's Note: An ability’s defense score is essentially its average roll. Requiring the rolling side to beat the opposing defense allows contests to be settled more quickly, eliminates the possibility of ties, and allows the game to be run with players doing all of the rolling if they so choose, since the odds of success are the same no matter which side rolls.
If there are situational factors that make a save significantly easier or harder, the referee may grant the roll advantage or disadvantage. If a roll has advantage, roll 2d20 and use the better of the two dice. If it has disadvantage, roll 2d20 and use the worse of the two dice.
Designer's Note: The referee is of course free to impose positive or negative modifiers rather than use the advantage system, but most players seem to enjoy it and it simplifies the math.