Certain cards and effects can negate the summon of a monster(s). These effects are activated "when a monster(s) would be summoned:". This is a point in time before the monster is successfully summoned; before the monster is face-up on the field.
Only cards or effects which state that they can be activated "when a monster(s) would be summoned:" may be activated as Chain Link 1 at this time. After this chain has been started however, standard chain rules apply. Additionally, an effect which negates a summon must be activated as Chain Link 1, and cannot be activated higher in the chain. Once this chain resolves, play advances, and it is no longer "when a monster(s) would be summoned:", so a second card or effect that negates a summon cannot be activated in a second chain.
Cards may specify that they may only be activated when a specific type of summon is being performed, whether this be a Normal Summon or Special Summon in general, or more specific such as a Fusion Summon, Synchro Summon, Normal Summon, Flip Summon, etc.
If a summon is negated, the monster is not considered to have ever been on the field, and has not been successfully summoned. Effects such as that of "Sangan" (which require the monster to be sent from the field to the Graveyard) will not activate, and Special Summon Monsters, which must be properly summoned first before being Special Summoned by other means, cannot be Special Summoned from the Graveyard with an effect such as "Monster Reborn". For example, if the Special Summon of "Black Luster Soldier - Envoy of the Beginning" is negated by "Solemn Warning", it cannot be Special Summoned from the Graveyard, as it has not yet been successfully summoned correctly.
As per the basic rules of dueling, cards and effects cannot be activated whilst a chain is resolving, as you cannot interrupt a resolving chain. As a result, if a chain is being resolved at the point in time a monster "would" be summoned, an effect that negates this summon cannot be activated.
"'Immediately After this effect resolves…'"
Some effects allow the player to perform a summon immediately after that effect resolves, outside of its Chain Link, such as "Urgent Tuning", "Formula Synchron", and "Ultimate Offering". In these cases, as long as the card or effect is Chain Link 1, that summon may be negated, as activating a card or effect at the point in time when the monster "would" be summoned would not be interrupting a resolving chain. If the effect is Chain Link 2 or higher, then a chain is still resolving, and this resolving chain cannot be interrupted.
Activated effects which Special Summon at a later time
Some activated effects also perform a Special Summon at a later time, after they have resolved, such as "Memory of an Adversary" and "W Nebula Meteorite". It is also not possible to activate an effect that would negate the Special Summon being performed by these kinds of effects, as you cannot interrupt an effect that is only being applied at one particular moment in time. Again, this is a general game mechanic not specific to summons.
- Player A attempts to Special Summon "Cyber Dragon" from their hand. This effect does not use the chain, so a card or effect that activates when this monster "would" (such as "Black Horn of Heaven") be summoned may be activated.
- Player A resolves the effect of "Polymerization", performing a Fusion Summon. At the point in time when the monster "would" be Fusion Summoned, the effect of "Polymerization" is still resolving, so the resolving chain cannot be interrupted with the activation of a new card or effect.
- Player A attempts to Xyz Summon "Gem-Knight Pearl". Player B activates the effect of "Thunder King Rai-Oh" as Chain Link 1. Player A cannot chain the effect of "Stardust Dragon", as "Gem-Knight Pearl" has not yet been successfully summoned, and is thus not on the field.