The only target of Vengeful Archon’s ability is the player it may deal damage to. You choose that target as you activate the ability, not at the time it prevents damage.

If the targeted player is an illegal target by the time the ability resolves, the entire ability is countered. No damage will be prevented.

Vengeful Archon’s ability doesn’t have you choose a source of damage. It will apply to the next X damage you would be dealt that turn, regardless of where that damage comes from. It doesn’t matter whether the damage would all be dealt at the same time. For example, if X is 4 and you’d be dealt 3 damage by Lightning Bolt, that 3 damage is prevented and Vengeful Archon deals 3 damage to the targeted player. The prevention effect will still apply to the next 1 damage you’d be dealt that turn.

The effect of Vengeful Archon’s ability is not a redirection effect. If it prevents damage, Vengeful Archon (not the source of that damage) deals damage to the targeted player as part of that prevention effect. Vengeful Archon is the source of the new damage, so the characteristics of the original source (such as its color, or whether it had lifelink or deathtouch) don’t affect this damage. The new damage is not combat damage, even if the prevented damage was. Since you control the source of the new damage, if you targeted an opponent with Vengeful Archon’s ability, you may have Vengeful Archon deal its damage to a planeswalker that opponent controls.

Whether the targeted player is still a legal target is no longer checked after Vengeful Archon’s ability resolves. For example, if a player targeted by Vengeful Archon’s ability puts Leyline of Sanctity (which says “You can’t be the target of spells or abilities your opponents control") onto the battlefield after the ability resolves but before it prevents damage, the ability will still prevent damage and still deal damage to that player. If Vengeful Archon can’t deal damage to the targeted player (because the player is no longer in the game in a multiplayer game, for example), it will still prevent damage. It just won’t deal any damage itself.

If Vengeful Archon’s ability doesn’t prevent any damage (perhaps because a different prevention effect is applied to the damage that would be dealt to you, or because the damage is unpreventable), Vengeful Archon won’t deal any damage itself.

If Vengeful Archon’s ability prevents damage, Vengeful Archon deals its damage immediately afterward as part of that same prevention effect. This happens before state-based actions are checked, and before any player can cast spells or activate abilities. If a spell or ability would have caused that damage to be dealt, this happens before that spell or ability resumes its resolution.

If you would be dealt combat damage by multiple attacking creatures, you choose which of that damage to prevent. For example, if X is 3 and you’d be dealt combat damage by a 1/3 Scroll Thief and a 3/5 Siege Mastodon, you might choose to prevent 1 damage from the Scroll Thief and 2 damage from the Siege Mastodon. You don’t decide until the point at which the creatures would deal their damage.

If the amount of damage that would be dealt to you is in excess of the amount of damage that Vengeful Archon’s ability would prevent, that source deals its excess damage to you at the same time that the rest of it is prevented. Then Vengeful Archon deals its damage.

If multiple effects modify how damage will be dealt, the player who would be dealt damage chooses the order to apply the effects. For example, say you’ve activated Vengeful Archon’s ability with X = 3 targeting Player A, and you’ve activated Vengeful Archon’s ability with X = 1 targeting Player B. You’re dealt 2 damage. You can apply the first prevention effect (dealing 2 damage to Player A), or you can apply the second prevention effect (dealing 1 damage to Player B) followed by the first prevention effect (dealing 1 damage to Player A). The unused portions of the prevention effects remain.