Resting

Short Rest

RAW:

A short rest is a period of downtime, at least 1 hour , during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
A character can spend one or more Hit Dice at the end of a short rest, up to the character's maximum number of Hit Dice, which is equal to the character's level. For each Hit Die spent in this way, the player rolls the die and adds the character's Constitution modifier to it. The character regains hit points equal to the total (minimum of 0). The player can decide to spend an additional Hit Die after each roll. A character regains some spent Hit Dice upon finishing a long rest, as explained below.

Activities during a short rest

Class features that explicitly state they happen during short rests (e.g. Song of Rest) can be safely performed during a short rest, but if you are:

• Traveling, patrolling, exploring, or investigating
• Making most skill checks
• Casting spells

then you are probably not resting.

Retroactive short rests

If the party has not taken any non-rest actions for significant time, a short rest can be called retroactively at DM discretion.

Long Rest

RAW:

A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch. If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.
At the end of a long rest, a character regains all lost hit points. The character also regains spent Hit Dice, up to a number of dice equal to half of the character's total number of them (minimum of one die). For example, if a character has eight Hit Dice, he or she can regain four spent Hit Dice upon finishing a long rest.
A character can't benefit from more than one long rest in a 24-hour period, and a character must have at least 1 hit point at the start of the rest to gain its benefits.

Long Rest and Armor

The rules about Sleeping in Armor from Xanathar's Guide to Everything will be used:

Sleeping in light armor has no adverse effect on the wearer, but sleeping in medium or heavy armor makes it difficult to recover fully during a long rest.
When you finish a long rest during which you slept in medium or heavy armor, you regain only one quarter of your spent Hit Dice (minimum of one die). If you have any levels of exhaustion, the rest doesn’t reduce your exhaustion level.

Note that there's a distinction between sleeping and resting; some races don't sleep, so this isn't applicable in their case.

Interrupting Long Rests

RAW:

• A long rest is (at least) 8 hours of downtime, during which at least 6 hours is spent sleeping. The other two hours can be spent standing watching, talking, eating, and similar light activities. An interruption of a full hour (combat, fleeing, other adventuring stuff) means the rest is spoiled and would have to start again.

Some races have exceptions to these outlines. Ol'Harb, for example, can complete a long rest in 4 hours if he spends it appropriately ("in an inactive, motionless state, during which you retain consciousness.")