Calculating Skills
Your skills are calculated based off your Ability Score Modifiers. The Ability Score Modifier that a skill uses are listed next to the skill. This modifier is the bonus applied to the 1d20 roll you make for that skill check.
Skill Proficiency
What does it mean to be proficient in a skill? It indicates extra time and effort to learn and practice that particular skill. When you are proficient in a skill, you add your Proficiency Score to that skill. Some skills you can attempt untrained but take disadvantage while others you must be proficient in to even attempt to use.
Important:
You can only be proficient in a number of skills equal to your Proficiency Score or Intelligence Modifier, whichever is higher.
Passive Skills
A Passive Perception is 10 + your Perception bonus, not including bonus dice. This represents your unconscious awareness at all times when you are not actively seeking something with your Perception. A Passive Perception becomes a DC for others to use as a marker for success or failure, where making equal to or higher than is a success.
In fact, all skills technically have this passive form, though some are not used often if ever, such as Passive History. These are called Passive [Insert Skill Name Here], such as Passive Athletics or Passive Insight. And, like Passive Perception, it is 10 + the skill bonus not including extra dice.
When do you use Passive Skills and when do you roll the skill? There are various ideas and rules. The most practical is whoever is actively performing the action rolls and compares it to the passive skill of all others. Such as a creature is inflicting the Grabbed condition. While they are holding you, when it is your turn in the encounter, you are actively trying to remove the condition, thus it is your Acrobatics or Athletics against the creatures Passive Athletics.
The same applies to Perception and Stealth and Deception and Insight. If you are actively searching, you roll against Passive Stealth. But if you are actively sneaking, you roll against Passive Perception. If you are lying, it is a roll against Passive Insight. But if you do not trust what someone else is saying, it is a check against their Passive Deception.
That rule holds until both creatures are active. If you are actively lying to someone who distrusts you and is actively trying the roll Insight, you might increase the creature’s Passive Insight, inflict disadvantage to your roll, or just compare two rolls.
Taking 10 or 20
To Take 10 means you forgo a 1d20 roll and instead treat the check as if you had rolled a 10 on the 1d20. Some traits allow you to do this and in some situations the GM might decide it is reasonable. Similar to Taking 10, Taking 20 is treated as if you had rolled a 20. However, choosing to Take 20 can never grant you a critical success.
Armor Check Penalty
If a skill includes the Armor Check Penalty, namely Acrobatics, Athletics, Stealth, and Thievery, then it is impacted by restrictive movement from certain armor. Any check penalty that you have from your armor gets added into these skill checks. Then range from 0 down into the negative numbers, reducing your skill bonus.