Criteria

OR

The OR criteria is quite powerful in creating advanced conditions.
Normally, when combining criteria, it means you need to meet all the criteria.
However, utilizing the OR criteria, it allows you to add other criteria as its children, and only checks if you meet one of the children criteria to satisfy the parent OR criteria.

criteria:
  - type: OR
    children:
    - type: PERMISSION
      permission: 'some.permission'
    - type: PERMISSION
      option: 'other.permission'

PERMISSION

The permission criteria will most likely be the most used criteria.
You can define any permission for a scoreboard's criteria.

criteria:
  - type: PERMISSION
  permission: 'example.permission'
  default: true

The permission specified is the required permission the player must have to meet the criteria.
default is optional, and defaults to false if not specified. If default is true, then the criteria will be met if even if they don't have the permission. If default is true, you must negate the permission from the player if you don't want them to meet the criteria.

EVENT

The event criteria is another powerful criteria.
This allows you to display scoreboards when specific events are triggered!
The most simple examples are the join and combat scoreboards.

criteria:
  - type: EVENT
    event: 'org.bukkit.event.player.PlayerJoinEvent'
    event-player: 'getPlayer'
    stay-seconds: 6

event is the event the criteria is listening for. You must use the full package of the event. You may use any plugin's custom events as well.
event-player is the event's method that will return the player.
stay-seconds determines how long the scoreboard will stay up.
Above is an example of a criteria listening for the PlayerJoinEvent.
What happens is, the player will meet this criteria once they join the server, and it will stay up for 6 seconds.

EVENT variables

You can also use variables to get different information from the event!
The syntax for an event variable is {event.<eventName>.<method>}.

You can call multiple methods.
For example: {event.EntityDamageByEntityEvent.getDamager.getName}
getDamager and getName are both methods.

Furthermore, if the method you input returns null, you can include a value to fall back to.
Syntax: {event.<eventName>.<method>:[fallback]}
Example: {event.PlayerJoinEvent.getPlayer.getName:%player_name%}
In the example, if getPlayer.getName returns null, it will display %player_name% instead (which is a PlaceholderAPI placeholder).

The combat scoreboard example showcases some variables being used.