Special Summon (特殊召喚 Tokushu Shōkan) is any Summon other than Normal Summon or Flip Summon. They usually occur due to a card effect or Summoning condition. Unless the effect indicates otherwise, the Special Summoned monster is placed directly onto a Monster Card Zone in face-up Attack Position or face-up Defense Position.
Special Summons do not take up the player's Normal Summon/Set for the turn, and there is no restriction on the number of times they can be performed in a turn. Special Summoning does not require Tributes for high-Level monsters like Normal Summons and Sets do, unless otherwise specified by a card.
Special Summon types
There are two different types of Special Summon. The type indicates which cards and effects can respond to or negate the Special Summon. Cards printed with Problem-Solving Card Text can easily have their method of Special Summon determined: effects that contain a colon or semicolon are Special Summons due to a card effect (and thus start a Chain), and cards that don't are built-in Special Summons.
Built-in Special Summons
Built-in Special Summons (also unofficially known as Inherent Special Summons) are Special Summons that are done by a Summoning Procedure either by its own Summoning mechanics (Synchro Summon, Xyz Summon), a Summoning condition or by an Effect without Classification that allows it to be Summoned that way. Generally, these are performed when the game state is open during your Main Phase (for example, "Cyber Dragon", "Solar Wind Jammer", "Ghost Ship", "Exodius the Ultimate Forbidden Lord", etc.). This type of Special Summon behaves similarly to Normal Summons and Flip Summons. Contact Fusions, Synchro Summons and Xyz Summons (except Rank-Ups) fall into this group of Special Summons.
Following the Problem-Solving Card Text, monsters' card text will specify in parenthesis "()" the location from which they can be Summoned, excluding Synchro and Xyz Summons. Monsters whose card texts do not specify the location from where they are Summoned are always Summoned from the hand if they are used in the Main Deck, and from the Extra Deck if they are used there.
These Special Summons are the only Special Summons that have a Summon negation window. As such, they can be negated by any card that can negate Special Summons, including "Solemn Warning", "Solemn Judgment", and "Black Horn of Heaven". Cards such as "Divine Wrath" cannot negate this type of Special Summon, because these Summoning Procedures do not activate nor are they card effects (although some of them are allowed by an Effect without Classification).
To go into more detail with Effects without Classification, Summoning conditions and Built-In Summons, if the monster can be Normal Summoned/Set with a Built-In Summon, then that is done by an Effect without Classification. Normal Summoning/Setting is the primary to place it on the field; therefore, any "extra" ways to Summon them are considered effects that allow them to be Summoned differently. The effect is applied on the monster in the location that it can be Special Summoned from, and for cards printed in PSCT, the location is in parenthesis "()". That effect allows the monster to be Special Summoned using the Summoning Procedure described on the card text from the specific area.
For example, "Grapha, Dragon Lord of Dark World" can be Normal Summoned/Set as its primary way of Summoning it; therefore any Extra Summons are done by an Effect without Classification. It is applied on Grapha in the Graveyard, and it allows Grapha to be Summoned using the Summoning Procedure of returning a "Dark World" monster to the hand and Special Summoning it from the Graveyard. If "Grapha, Dragon Lord of Dark World" is destroyed by battle with "Dark Ruler Ha Des", then Ha Des will negate Grapha's effect without classification in the Graveyard; therefore, Grapha cannot use the Summoning Procedure at all to Special Summon itself.
However, if the monster is a Special-Summon only monster (including "Nomi" monsters), then its Built-In Summon is a Summoning condition. Since it cannot be Normal Summoned/Set, it must be Special Summoned on the way written on the card (or must first be Summoned) as the primary way to Summon it. The difference between these two methods of Built-In Summons is shown in the card rulings of "Vampire Grace".
Special Summon with a card effect
Also called Effect Summons, these are any Special Summon that results or occurs at the resolution of an activated card effect that Special Summons (e.g. "Monster Reborn" and "Gorz the Emissary of Darkness"). Fusion Summons, Ritual Summons and Rank-Ups fall under this group of Special Summons.
These Special Summons cannot be negated, but they can be stopped by negating the initial effect that caused the Special Summon. Cards such as "Black Horn of Heaven" and "Thunder King Rai-Oh" cannot negate this Special Summon, because it occurs during resolution of a card effect, and you cannot activate anything (and start and resolve a new Chain) when resolving another card or effect; there cannot be a Chain within another Chain. However, cards such as "Divine Wrath" and "Dark Bribe" can negate the activation of a monster effect or Spell/Trap Card that would Special Summon a monster, respectively. "Solemn Judgement" can negate either a Spell/Trap effect (including one that would Special Summon a monster) or a built-in Special Summon; however, it cannot negate monster effects that Special Summon. "Solemn Warning" can negate any effect that Special Summons, as well as built-in Special Summons (if it is a Spell or Trap card, "Solemn Warning" can only negate it if it is the initial activation of the card as it cannot negate activation of effects).
Summon response window
The "Summon Response Window" is an unofficial term, used to describe the window that responds to a successful Summon (an action that does NOT start a Chain; go see Fast Effect Timing). Cards and effects that can be activated in response to a successful Special Summon, such as "Torrential Tribute" and "Bottomless Trap Hole", can be activated after any of these Special Summons, regardless of whether it is built-in or because of a resolved effect. However, they cannot be activated if a monster is Special Summoned by a card effect that is Chain Link 2 or higher, unless the activation of all other cards and effects that would resolve after it are negated. This is because cards and effects cannot be activated in the middle of a resolving Chain, and whatever effect resolves last (as Chain Link 1) would cause these cards to miss the timing.
Furthermore, if the card effect that Special Summoned the monster did something else immediately after Special Summoning the monster (e.g. "Magical Dimension", "Magical Dimension", "Drill Warrior", "Telekinetic Power Well", "Trojan Gladiator Beast", etc.), then this window does not occur and any cards that can be activated in response to a Summon miss the timing.