These are the most common terms used in Fate.
Mana/Od: Both Mana and Od are two types of magical energy. Mana is produced by The World and can be gathered from the air and earth, but particularly coalesces in streams call Ley Lines. Od is the magical energy taken from the World and refined by the Magic Circuits in the body.
Holy Grail: The legendary holy cup. Grants wishes by opening a hole to The Root. This allows access to nearly unlimited Mana. The Greater Grail is a semi-sentient being who chooses Masters and Servants. The Lesser Grail is a physical object that absorbs the magical energy of fallen Servants. Once enough Servants are absorbed, the Lesser Grail can access The Root from the physical world.
Holy Grail War: The fight over the Holy Grail. Typically comes with with its own set of rules. The Fuyuki Grail War began in the 1800s and occurs every 60 years. It began as an attempt to access The Root.
Master: A mage who has entered into a contract with a Servant.
Heroic Spirit: A copy of humanity's greatest heroes that are kept in the Throne of Heroes. These beings are a combination of the actual person and all myths/legends/feats attributed to them.
Servant: A copy of a Heroic Spirit from the Throne of Heroes bound to the physical world by a Master.
Class: The archetype vessel into which Heroic Spirit can be summoned. These archetypes make it easier for the Holy Grail manifest Heroic Spirits as Servants in the physical world. The main 7 classes in the Fuyuki Grail War are Saber, Lancer, Archer, Caster, Assassin, and Berserker. The classes of Shielder, Ruler, Avenger, and Saver have also been observed in unique cases. Command Seal/Spell: A magical mark that can be used to compel Servants into action. They are made of pure mana, so they can be used for other purposes as well. Masters are typically given three Command Seals.
Noble Phantasm: A weapon or ability used by Heroic Spirits. These are usually symbols of their renown and represent a Heroic Spirit's unique strengths and abilities.
Mystic Code: A weapon or magical item used by mages to augment their magical or combat abilities.
Familiar: A magical creature that obeys the will of its master. Typically takes the form of a small animal, but technically includes Servants, golems, and puppets.
The Throne of Heroes: A location outside of time and space where Alaya stores Heroic Spirits. These beings are usually used as agents of The Counter Force. However, mages have broken into this place to use Heroic Spirits as Servants in Holy Grail Wars.
Homunculi: Artificial human-like beings most commonly produced by the Einzbern family. These beings have no souls but are adept at producing magical energy and following their programming.
The Mage's Association: The most prominent group of mages in the world. From their headquarters in the Clocktower of London, these mages seek to teach, regulate, and keep magic a secret from non-mages.
Magic/Thaumaturgy: There are two types of magecraft: Thaumaturgy and Magic. Magic is sometimes called True Magic to separate it from the general term of magic. Thaumaturgy is any magecraft that could be accomplished by sufficiently advanced science, but is done through supernatural means instead. For example, lead can be transmuted to gold through transmutation magic, but can also be transformed through nuclear fission. True Magic is something that cannot be done through science. As science and magecraft advances, the classification of certain spells can change from Magic to Thaumaturgy and vice versa. Currently, there are only 5 True Magics, which are listed below. First Magic: Unknown. Could be the concept known as the "Denial of Nothingness", which allows for the creation of matter from nothing.
Second Magic: Known as the "Operation of Parallel Worlds". Allows a user to shift into parallel universes. This is the reason why all Fate works can be considered canon. Kischur Zelretch Schweinorg is its current user.
Third Magic: Known as the "Materialization of the Soul". Grants immortality and resurrection abilities by removing the soul from the cycle of The Root. The Third Magic was originally achieved by the Einzbern family, but its secret was lost one thousand years ago. For the next eight centuries, the Einzberns sought to recover it until finally coming up with the concept of the Holy Grail War.
Fourth Magic: Details are unknown, but mages agree it definitely exists.
Fifth Magic: Time Travel. Though, because this can be accomplished using the Second Magic, there may be other elements yet unknown. Aoko Aozaki is its current user.
The Root/Swirl of the Root/Akasha: The source of all Mana and human souls. Most mages endeavor to access the Root for purposes of immortality, power, or renown. When a human dies, their soul is returned to the Root. When a human is born, a soul is taken from the Root. Because the Root's power is spread across humans more or less equally, the less humans there are, the more magical potential each will hold. In the past, magic was commonplace. However, it has all but faded in the present day. magic
Alaya: A semi-sentient being who is the collective conscience of humanity. Alaya was once a part of Gaia, but grew too powerful. Alaya collects the best of humanity as Heroic Spirits in the Throne of Heroes. They are used to defend humanity as actors of the Counter Force.
Gaia: A semi-sentient being who acts as the conscious will of the planet Earth. Gaia uses magical beasts to defend nature and animals. These creatures are the agents of the Counter Force. Supernatural acts of magic require energy called Mana to maintain or will be destroyed by Gaia. Gaia frequently comes into conflict with Alaya.
The Church: A Christian group that seeks to collect holy artifacts, deal with sinful abominations such as Vampires, and control believers. Frequently comes into conflict with The Mage's Association.
The Counterforce: A term used to describe actions taken by Gaia or Alaya to defend their respective domain. If one grows too strong and threatens the other's safety, the defendant will summon certain agents such as Heroic Spirits or magical beasts. Gaia and Alaya endlessly counter each other's efforts and keep the word in balance. In the past few thousand years, Alaya has grown significantly in power.
Dead Apostles: Humans that have become vampires, gaining a zombie like hunger for human flesh. One of the main targets of The Church.