Summoning

Summoning allows players to receive additional Servants, Craft Essences, Experience Cards, and Status Up Cards. There are currently two ways of summoning: Saint Quartz summoning and Friend Point summoning.

Saint Quartz Summoning
Saint Quartz summoning is used to summon Servants and Craft Essences of 3-5★ rarity. Quartz summoning requires either 3 Saint Quartz or 1 per individual summon, or 30 Saint Quartz to summon 10 cards. When performing a 10x summon (a.k.a. 10-roll), you are guaranteed to summon at least one 3-5★ Servant and one 4-5★ Servant or Craft Essence. You cannot perform a 10-roll with 10 Summon Tickets. The Saint Quartz summoning pool that is always available is known as "story summoning." Sometimes When a 4-5★ Servant is summoned, the summoning animation's blue ring will turn gold.

Some Servants are only available in the Quartz summoning pool after completing main story chapters.
 * Artoria Pendragon (Alter) & Cu Chulainn (Caster) (Fuyuki)
 * Gilles de Rais (Orléans)
 * Nero Claudius (Septem)
 * Medea (Lily) (Okeanos)
 * Nikola Tesla & Artoria Pendragon (Lancer Alter) (London)
 * Li Shuwen (Lancer), Medb, Thomas Edison & Cu Chulainn (Alter) (E Pluribus Unum)
 * Gawain, Bedivere, Tristan & Artoria Pendragon (Lancer) (Camelot)
 * Jaguar Man, Quetzalcoatl & Gorgon (Babylonia)
 * EMIYA (Alter) (Shinjuku)

Here is the in-game stated probability of summoning different rarities of Servants and Craft Essences with Summon Tickets/Saint Quartz:

Limited Summoning Campaigns
There may be additional splash pages for Saint Quartz summoning during ongoing events or promotions. For any of these summonings, the Servants and Craft Essences available may be different. To check the list of cards in the current summoning pool, tap the Summon Details (召喚詳細) button in the lower left. This includes Servants that are normally unlocked through the story; limited summoning campaigns may temporarily make them available even if you have not progressed that far yet.

Limited summoning campaigns usually offer a "rate up" for a specified number of Servants; this does not change the summoning probabilities in the table above!!! The chance of summoning a 5★ Servant is still 1%, but, if you were to summon a 5★, there is a higher chance it will be a rate-up Servant.

Some Servants can only be summoned during limited summoning campaigns - these Servants are commonly known as "Limited Servants." To date, no plans have been announced to add them to the permanent summoning pool, but they may appear again in limited summoning campaigns in the future.
 * Gilgamesh
 * Sakata Kintoki
 * Okita Souji
 * Scathach
 * Mysterious Heroine X
 * Brynhildr
 * Nero Claudius (Bride)
 * Ryougi Shiki (Saber)
 * Amakusa Shirou
 * Edmond Dantes
 * Jeanne d'Arc (Alter)
 * Iskandar
 * Shuten Douji
 * Minamoto no Yorimitsu
 * Leonardo Da Vinci
 * Tamamo no Mae (Lancer)
 * Artoria Pendragon (Archer)
 * Marie Antoinette (Caster)
 * Anne Bonny & Mary Read (Archer)
 * Mordred (Rider)
 * Kiyohime (Lancer)
 * Saint Martha (Ruler)
 * Illyasviel von Einzbern
 * Cleopatra
 * Ishtar
 * Merlin
 * Miyamoto Musashi
 * King Hassan
 * Mysterious Heroine X (Alter)
 * James Moriarty
 * Arthur Pendragon (Prototype)
 * Hijikata Toshizou
 * Meltlilith
 * Passionlip
 * Sessyoin Kiara

Sometimes Servants are released before their main story location has been released. These Servants are temporarily considered "limited Servants," but will be added to the permanent summoning pool at a later date.
 * Currently no servant

Some limited summoning campaigns (Example, Guaranteed 5 Summoning) require the use of paid Saint Quartz. When attempting a Saint Quartz summoning, underneath your total number of Saint Quartz, the number listed on the left is your paid Saint Quartz, while the number listed on the right is your free Saint Quartz.

Friend Point Summoning
Friend Point summoning can summon any type of card at 3★ or lower rarity. As of 7M Downloads Campaign update, Each player receives 10x free summon per day at 00:00 JST. Subsequent summons require 200 FP each and 2000 FP for each 10x. Friend Points can be commonly obtained from Login Rewards, using other players' Servants as support, and other players using your Servant as support.

Starter Summoning
During the opening tutorial, there is a forced Saint Quartz summoning where players get their first Servant. As of E Pluribus Unum Chapter Release, the players are guaranteed to get a 4★ Servant (from below table) and a 4★ Craft Essence.

Trivia

 * For Saint Quartz summoning, the chance to summon 3★ Servants and 3★ Craft Essences is now equally 40% with the release of Okeanos. Previously, the chance of summoning 3★ Craft Essences was much higher than that of 3★ Servants.
 * With the release of E Pluribus Unum, the Tutorial Summoning pool was expanded to include many 4★ Servants. It also introduced the special 10-card summon that guarantees a 4★ Servant and a 4★ Craft Essence.
 * With the onset Fate/Grand Order's 1st Anniversary, the amount of Saint Quartz used to summon was reduced from 4 to 3.
 * A solution for those who mourn for the failure of getting their favorite limited servants is to play in the Chinese server, available in Appstore. The Chinese server was delegated by a Chinese company to operate in China beginning in October 2016, meaning that the entire schedule (including the limited summoning campaigns, events and event servants, and other features) is about a year later than the Japanese server. Unless you read Japanese, there would not be much of a difference in the playing the game. The summoning cost is an exception, as it begins as 3 per summon in the Chinese server. You would need an account in the Chinese Appstore for this, which should not be difficult to register.
 * A quick general walk through of all the SSR of this game (with music to go with their NPs) for those in need of finding Waifu: (the description in Chinese is only half a minute. you can skip that.)
 * Volume 1: http://www.bilibili.com/video/av8265702/
 * Volume 2: http://www.bilibili.com/video/av8408120/