Ability Checks
Roll a d20, add the relevant ability modifier, and add your proficiency bonus if you're proficient in the skill or tool being used. Meet or beat the Difficulty Class (DC) to succeed. Typical DCs: 5 very easy, 10 easy, 15 medium, 20 hard, 25 very hard, 30 nearly impossible.
Working Together & Group Checks
A character can help another's check, granting advantage — but only if the helper could attempt the task alone. For group checks (sneaking past guards together), everyone rolls; the group succeeds if at least half succeed.
Saving Throws
A saving throw resists a spell, trap, poison, or similar threat: d20 + ability modifier + proficiency if you're proficient in that save (each class grants two). The six saves map to the six abilities — Strength (being moved), Dexterity (dodging), Constitution (enduring), Intelligence (disbelieving illusions), Wisdom (resisting charm/fear), Charisma (asserting identity against possession).
Passive Checks
A passive check is 10 + all normal modifiers — used when the DM wants to resolve something secretly or as a floor for repeated attempts. Passive Perception is the classic: it determines whether you notice ambushers without rolling.