There is no restriction on the number of times a player can Special Summon in a turn—Special Summons do not take up the player's Normal Summon/Set for the 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 Summon
Built-in Special Summons (also unofficially known as inherent Special Summons) are Special Summons that are done by a game mechanic (Synchro Summon, Xyz Summon, Pendulum Summon), the monsters' own Summoning condition (Nomi, Semi-Nomi) or a Summon monster effect.
Generally, these are performed when the game state is open during your Main Phase (e.g. "Cyber Dragon", "Stardust Dragon", "Number 39: Utopia", "Exodius the Ultimate Forbidden Lord"). They are never performed as part of an effect resolution; effects that would otherwise do this instead perform the Summon immediately after they resolve (e.g. "Formula Synchron", "Advanced Heraldry Art"), or are actually Special Summons with a card effect that treat the Summon as a Built-in Special Summon, like Xyz Summon (e.g. "Rank-Up-Magic").
For cards with Problem-Solving Card Text, monsters with Summon monster effects or Summoning conditions that Special Summon themselves this way specify in parenthesis "()" the location from which they can be Special Summoned.
There is a difference between Summoning a monster with a Summon monster effect and with a Summoning conditions. If the monster can be Normal Summoned/Set, then its built-in Special Summon is done by a Summon monster effect, which is a type of Unclassified Effect. Conversely, if the monster is a Special Summon monster, then its built-in Special Summon is a Summoning condition. This distinction causes subtle differences; for example, the effect of "Vampire Grace" can be activated when "Goka, the Pyre of Malice" is Special Summoned by its own Summon monster effect, but not when "Vampire Genesis" is Special Summoned by its own Summon condition.
Built-in Special Summons are the only kind of Special Summons that can be negated. When a player attempts to perform a built-in Special Summon, the Summon negation window opens, providing both players with the opportunity to negate the Summon. However, as performing a built-in Special Summon does not go on a Chain, cards such as "Divine Wrath" cannot be Chained to the attempted Summon.
Special Summon with a card effect
Special Summons with a card effect are any Special Summon that occurs during the resolution of an activated card effect (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. However, they can be prevented from occurring in the first place by negating the effect that would perform the Special Summon; 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.
While they are rare, Special Summons as the result of an activated card or effect that do not occur during that effect's resolution, except Synchro and Xyz Summons, also fall into this category. Cards such as "Revival Jam" and "Memory of an Adversary" can perform this kind of Special Summon.
Summon response window
The Summon response window is an unofficial term used to describe the timing in which cards and effects can be activated in response to a successful Summon. 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 with a card 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 whichever 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 performed another action during its resolution 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.