Targeting in Yu-Gi-Oh! definition selecting a certain monster in a certain area for an effect of a card. If a card effect targets a monster, it must happen at the point of activation before a response can be asked. If a card does NOT target, then it will choose a card for its effect when it resolves. Targeting is always in the form of an effect, never a cost. There are cards that seem like they target, however do not.
For a card effect to target another card, it must designate 1 region and a specific number of monsters.
Monster Reborn
Type: Spell (Normal)
Text: Target 1 monster in either player's Graveyard; Special Summon it.
Monster Reborn aims for the graveyard region and targets
1 monster from that area, therefore Monster Reborn targets. Before someone makes a response, the player who activated monster reborn must declare a target first, then the opposing of current player can respond to it.
Ex.
Player 1: Activate Monster Reborn
then targets Slifer, the Sky Dragon in his/her grave.
Player 2: Response Activates Solemn Warning and pays 2000 LP
Chain Resolves
Chain Link 2: Solemn Warning negates Monster Reborn
Chain Link 1: Monster Reborn's chain no longer exists and ends at chain link 2
Card effects that card aim for card in multiple regions do NOT target. The card that will be selected for the effect will be declared at the resolution of the chain.
Inzektor Dragonfly
Attribute: Dark
Level: 3
Type:Insect/Effect
ATK/DEF: 1000/1800
Text: Once per turn: You can equip 1 "Inzektor" monster from your hand or Graveyard to this card. If an Equip Card(s) is sent to your Graveyard while equipped to this card (except during the Damage Step): You can Special Summon 1 "Inzektor" monster from your Deck, except "Inzektor Dragonfly". While this card is equipped to a monster, that monster's Level is increased by 3.
Since Inzektor Dragonfly can equip cards to it from hand OR grave, it does not target. When Inzektor Dragonfly activates its effect, the controller of Inzektor Dragonfly does not declare its target until the resolution of the chain.
Ex.
Player 1: Activates Inzektor Dragonfly's effect.
Player 2: Response, discards D.D. Crow and targets Inzektor Hornet in the grave.
Chain Resolves
Chain Link 2: D.D. Crow banishes Inzektor Hornet from the grave.
Chain Link 1: Inzektor Dragonfly selects Inzektor Hornet from his/her hand and equips it.
Some cards sound like they would target, but truly they do not. There some patterns that will be used to explain why these cards do not target.
Ally of Justice Catastor
Attribute: Dark
Level: 5
Type: Machine/Synchro/Effect
ATK/DEF: 2200/1200
Text: At the start of the Damage Step, if this card battles a non-DARK monster: Destroy that monster immediately (without damage calculation).
It SOUNDS like Ally of Justice Catastor would target a non-Dark monster and utterly destroy it. Here's why this card does not target. Well in a nutshell, it doesn't designate a specific number of monsters to destroy. So ally of Justice Catastor does not target a non-Dark monster to destroy.
Bottomless Trap Hole
Type: Trap (Normal)
Text: Activate only when your opponent Summons a monster(s) with 1500 or more ATK. Destroy and remove from play the monster(s).
It would seem that Bottomless Trap Hole would target since it is commonly used to get rid of single cards, but the little (s) means that in any case multiple monsters with 1500 or more are special summoned at the same time, destroy them all. This card does not have a fixed number of targets (again...) and therefore does not target.
Card that designate cards in the hand or deck do not target (ex. Spirit Reaper, Mystic Tomato, ect.)
Fusion Cards, Ritual Cards, Synchro-ing support (ex. Formula Synchron, Urgent Tuning, etc.), and possible any XYZ-ing support do not target!
If I miss anything or if you have any questions or concerns, please do not hesitate to reply to this post. I will attend to it as soon as I can.
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Subject: What is "Targeting"? 
