When an effect targets, it chooses at the time of activation the specific card(s) that it will affect.
You must declare the target before players are allowed to chain to the effect. If another effect is chained to the original effect so that the target is not a legal target when resolving the original effect, then you do not get to "re-select" the target. The original effect resolves without effect (that is, it does nothing).
- Example: Player A activates "Offerings to the Doomed" and targets "Dark Magician". Player B chains "Book of Moon", targeting the same "Dark Magician". "Book of Moon" resolves first, and changes "Dark Magician" to face-down Defense Position. Then, when "Offerings to the Doomed" resolves, "Dark Magician" is not destroyed - "Offerings to the Doomed" can only target face-up monsters, and "Dark Magician" is face-down, so "Offerings to the Doomed" resolves without effect. You do not get to select a new monster to destroy.
- Example: Player A attacks with The Tricky. Player B activates Draining Shield. Draining Shield targets because it selects The Tricky's attack to negate. If Player A activates Trap Jammer, then Draining Shield resolves without effect and cannot be set back.
Equip Spell Cards always target the monster which is being equipped.
Cards like "Dark Hole", "Mirror Force", "Lightning Vortex" and "Raigeki" do not target, because the player does not specify which cards will be affected - rather, it destroys all monster, regardless of whether or not the monsters were on the field at activation. Likewise, "Fissure" and "Magical Dimension" do not target, because they select the monster to be destroyed when you are resolving the effect.
Effects that automatically choose what they are going to affect, like "Drillroid" or "D.D. Warrior Lady", usually do not target. Most Counter Trap Cards do not target for the same reason - most can only affect the effect directly beneath it in the Chain, so there is no selection, so the effect does not target.
If the card's text says "target", then the effect almost always targets. If the card's text says "select", then the effect usually targets, although there are some exceptions. When in doubt, check the card's rulings to see whether or not it targets.