The main idea behind these box templates is to give information that would be repeated (such as no AP or cooldown in the case of Passive ability) and also point out which attacks are nonbasic attack, or which effects are loss of life effects, so someone who is reading makes note of that.
Nonbasic attack
This is a nonbasic attack. This means it will not trigger some effects that specifically mention "basic attack" such as Rend and Swarm
Basic attack
This is a basic attack. This means it is on the Attack chain and will trigger effects that specifically mention "basic attack" such as Rend and Swarm.
Basic attack required
This ability requires a basic attack. This means it will not trigger from nonbasic attacks (and damage abilities) such as Charge and Stab which do not specifically mention "basic attack".
Successful attack required
This only requires a successful attack. This means it will even trigger from 0 damage, but not if it misses (for example, from being Blinded or from Block triggering).
Damage required
This needs to damage to be successful. This means it will not affect champions who are Impervious or otherwise don't take damage. (Note if another champion takes the damage, that champion will be affected instead.)
Damage ability
This is a damage ability. This means although it's on the attack chain, it does not trigger attack related abilities such as Dodge or Deflect, or basic attack abilities such as Enrage.
Activated ability
This is an activated ability. This means you can spend AP to use the ability on your turn and it may go on cooldown after being used.
Passive Ability
This is a passive ability. This means it does not need to be activated and works as long as the champion has it. It has no AP cost or cooldown.
Loss of life (loss of HP) effect
This uses loss of life (loss of HP). This means it will ignore any effect that reduces damage (such as Impervious) and not trigger any effect that triggers on damage (such as Enrage).