Fate/Kaleid/World info
Short version (for those familiar with the Fate series)
For those of you familiar with the Fate series, Fate/Kaleid is a spin-off alternate history version of the world where Iri was saved during the events of Fate/Zero, and the Holy Grail war of Fate/Stay Night never happens. All the same magic and family ties still exist, but the lack of the Holy Grail war has given certain people a chance at a much happier life, the Emiya/Einzbern family in particular.
Long Version
This is a world very much like Earth, but a society of mage craft has existed under secrecy for Millennia. Most mage craft in the world is held by the great mage families who pass down the secrets of magic from parent to child, however there are a few new mages with natural talent for magic that discover it on their own. It is usually the case that older mage families have more structured magic that is more versitile due to it having been perfected and passed down over generations, where as newer mage families are started by someone with great natural talent and large stores of mana, but their lack of structured knowledge greatly limits what they can do.
In this world, every few decades the Holy Grail appears and grants its power to the mage who wins a mage contest known as the Holy Grail war. Few people even in the Mage world know, but the holy grail being fought over is actually a creation of the Einzbern family, a family of powerful homunculi that are indistinguishable from humans and utilize powerful magic. They seek to obtain the holy grail themselves, but due to the grail's nature they are forced to invite 6 others into the contest in order to fight for the right to obtain it.
Those recognized by the grail are given the power to summon a heroic spirit, a great individual from any point in time who possessed magical or magically supported combat ability far exceeding that of a normal mage. This is how things normally work in this world anyway. However, in this alternate version of the world (for some spoiler free reasons,) the grail did not form and did not recognize any participant in the contest. As a result, instead of giving mages the ability to summon these heroic spirits, the spirits instead were summoned as cards that contained the essence of their spirits and magical energies and they began effecting ley lines that control the magical energy of the area around where the holy grail war was supposed to have taken place.
When the mages' association learned of this going on, they sent investigators out to find out what was going on, including some of their best combat mages. Once the investigators found out, defeated 2 of the heroic spirits, and then obtained the cards that represented their magical energies, the investigation was called off and the cards brought back for examination. The task of collecting the remaining cards was then passed off as a test to 2 hopeful students of the association who wished to be recognized and taken in to study directly under the mage association's grand master himself.
Mage's Association
Little canonical information is given about the Mage's association, but the association is located in London and features the most prestigious school of magecraft in the world. This school functions as a University would in non-magical society, while your preliminary studies in magecraft are expected to have come from your parents.
The Mage's Society seems to be divided into several factions vying for power and control, the Grand Church and the Mage's Association being the two most significant, and they seem to be acknowledged and approved of at least in part by the government of London as their head office is located in the Westminster Clock Tower.
The church is actually separate from the mage's association, and it is only recently that the two seem to have even as much as decided to tolerate one another. Up until about 200 years ago, the two were in a fight to the death with one another.
The Holy Grail War
The Holy Grail War is actually a summoning ritual meant to summon the Holy Grail. Three of the oldest and most powerful mage families act together in order to conduct the ritual, and 4 more mage families are permitted into it because a minimum of 6 sacrifices are needed to complete the ritual. The ritual takes the form of a mock battle, but it is in fact a battle to the death. Representatives from 7 mage families, 3 of witch are the 3 families who made the preparations for the ritual, fight to the death using summoned heroes of the past. The objective is really to eliminate the 6 other heroes, but it is not at all uncommon to attack the summoner themselves as they are a soft target by comparison to their summoned hero.
The way that the ritual is conducted is that the partially summoned holy grail is used as a catalyst to empower 7 summoners to summon 7 heroic spirits. Heroic spirits are the very strongest class of summoned spirits as they are the incarnations of powerful heroes from across time. It would normally be far beyond the ability of any magus to summon a heroic spirit, let alone command them or provide them with the energy necessary to give them physical form. However, with the partially summoned holy grail as a catalyst, it is all possible. Once the heroes are summoned, 6 of them must be sacrificed to the grail. The grail will then fully manifest and can be obtained as a prize by the summoner of the one surviving hero.
The preparations for this ritual conducted by the 3 great families presiding over the holy grail war are that a physical vessel must be prepared to become the grail, and a summoning circle must be prepared at a power spot of multiple very powerful lay-lines. The vessel of the grail is a Homunculus' heart that has been specially prepared with very powerful magic. The Einzbern family is considered to be the best in the entire magic world at creating Homunculi, and so they are the ones to prepare the vessel. The Toshaka family are a family that once specialized in summoning, and they also happened to control a territory sufficient for the summoning. The ritual could technically take place with only these 2 families, but in order to ensure the summoned heroes were better controlled and thus increase the odds of success in the ritual, the Makiri family who specialized in the use of familiars were invited in to cooperate with the ritual for their own guaranteed chance to participate in the ritual. They provided a command seal that gave the summoner increased control over their summoned hero as well as 3 chances to give their hero an order that absolutely cannot be refused.