L. Scott Johnson is right, but I'd like to add a bit more of an explanation.
Basically, it depends on Spell Speed
In yugioh, all spells, traps, and otherwise activated card effects have a Spell Speed. This property determines things like when the cards/effects can be activated, and when/if they can be activated in response to another activation.
Spell Speed 1 cards and effects may only be activated by the turn player during their Main Phase 1/2, and may only be activated as the first effect in the chain. They cannot be used to respond to other effects.
Spell Speed 2 cards and effects may being activated during either player's turn, during any phase (except during the damage step; this is more nuanced and touched upon in a bit), and at any point in the chain - provided that the conditions for their activation are met, and that the most recently activated card/effect is of Spell Speed 2 or less.
The only Spell Speed 3 cards are Counter Traps - these can be activated anytime their conditions for activation (typically quite restrictive) are met, unless explicitly forbidden by some other card effect.
You may hear some people refer to "Spell Speed 4"; this is just a player-invented term for cards like Super Polymerization which read "Neither player can activate cards or effects in response to this card's activation."
Normal, Continuous, Equip and Field spell cards are all inherently considered to have Spell Speed 1, as are all of their effects. Some of these, like Supply Squad, have effects that can activate during the opponent's turn and/or after the start of the chain, but these are generally triggered by something mandatory and are still considered to have Spell Speed 1.
Quick-Play Spell cards, along with Normal and Continuous Trap Cards, are all inherently considered to have Spell Speed 2, as are all of their effects. This is why, in the case of Eternal Soul, the "Once per turn" effect can still be activated during the opponent's turn - the effect is one of a continuous trap card, and so is inherently Spell Speed 2 (a Quick Effect). As another example, the effect of Unending Nightmare can be activated during either player's turn as well.
In the context of monster effects
The luxury of inferred Spell Speed unfortunately does not exist for monsters. We are, however, given very consistent text conventions to indicate the Spell Speed of monster effects. Monster effects will quite explicitly state whether an effect is Spell Speed 2 or not. The old convention, "During either player's turn," has been updated to "(Quick Effect)". If you see either of those phrases in the activation text of any card, the effect is Spell Speed 2. Otherwise, the effect is Spell Speed 1.
Damage Step sometimes makes things muddy
The Damage Step restricts what sorts of cards and effects are allowed to activate. Most obviously, cards that say "During the damage step", "During damage calculation", "When ___ is destroyed by battle", etc. are allowed to activate at that time. Outside of this, however, the rule of thumb is that a card/effect may not activate during the damage step unless:
- It is a counter-trap card
- Its activation is mandatory
- It negates the activation of a card/effect (Note Activation, not Effect of)
- It directly modifies the Attack/Defense of any monster(s)