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 game machine (Pendulum Summon), its own summoning condition (Synchro Summon, Xyz Summon) or an effect without classification (Summon monster effect) that allows it to be Summoned itself that way. Generally, these are performed when the game state is open during your Main Phase (for example, "Cyber Dragon", "Stardust Dragon", "Number 39: Utopia", "Exodius the Ultimate Forbidden Lord", etc.). This type of Special Summon behaves similarly to Normal Summons and Flip Summons.
Following the Problem-Solving Card Text, monsters' card text will specify in parenthesis "()" the location from which they can be Special Summoned (Built-in Special Summon), excluding Synchro, Xyz and Pendulum 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.
Note that there is a difference between Summoning a monster by an effect without classification or by Summoning conditions. If the monster can be Normal Summoned/Set, then its Built-In Special Summon by the way described in its text is considered done by an effect without classification. Normal Summoning/Setting here is the primary way to place the monster on the field; therefore, any "extra" ways written to Special Summon them are considered effects that allow them to be Summoned differently. However, if the monster is a Special Summon monster, then its Built-In Special 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".
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 and start a Chain.
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", "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.