5e Sparknotes

The chapters and headers listed here match those in the Player's Handbook (PHB). Please refer to the PHB for further info/clarification.

Chapter 7: Using Ability Scores

There are six attributes (important general usage in brackets per my opinion; each class has different requirements):

Ability Scores and Modifiers

A score of 10-11 is "average human," 20 is the maximum for player characters. Modifier is used much more than score.

Score Modifier
1 -5
2-3 -4
4-5 -3
6-7 -2
8-9 -1
10-11 +0
12-13 +1
14-15 +2
16-17 +3
18-19 +4
20-21 +5

Advantage and Disadvantage

Only affects d20 rolls. Advantage is "roll 2d20 use highest," disadvantage is "roll 2d20 use lowest." Neither effect stacks. You are either "at advantage/at disadvantage," or not. Advantage and disadvantage cancel each other out.

Proficiency Bonus

Only affects d20 rolls. Does not stack. You are either "proficient," or not.

Ability Checks

When attempting to do anything that has a chance to fail. Roll a d20 and add relevant bonuses. Players generally trigger these by saying things like "I try to hide" or "Can I examine X for traces of magic?".

Contests

Tug-of-war scenarios. Highest roll (with bonuses) wins. Ties generally result in neither side winning.

Skills

See "Using Each Ability" in the PHB for a breakdown.

Passive Checks

How good your character is when you're AFK. Passive scores are 10 + all relevant modifiers. Mainly used for passive Perception, for spotting stealthy enemies or hidden traps.

Working Together

One character proficient in a check can help another, giving them Advantage on their check.

Group Checks

Can be called for by the DM. If at least half the group succeeds, the group succeeds.

Saving Throws

Roll a d20 and add relevant bonuses. Used to avoid negative effects.

Chapter 8: Adventuring

Resting

Short Rest

1 hour rest (DM dependent). On completion, spend Hit Dice to heal. Some things recharge on short rest.

Long Rest

8 hour rest (DM dependent). 6 hours minimum sleeping, 2 hours maximum light activity. Can't benefit from more than one in a 24-hour period. If interrupted by more than an hour of strenuous activity, must be restarted.

Chapter 9: Combat

Turn-based, cycles through combatants in initiative order -- a round is one turn from each participant. A turn is action + bonus action + movement (all optional).

See this quick reference for the short and sweet version, or to refer to when playing. Alternatively, see PHB p189-198, this stuff is all pretty important.