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Note: due to the wide variety among Christian sects/denominations this section will mostly cover the early aspects of Christianity, up to and including the times of Luther. After that, separate entries will exist on the Roman Catholic church and the various other Protestant sects as people write them.
Christianity is the term for the various groups that follow the teachings of Jesus Christ. This is probably the only universal tenant, as Christianity has various branches, denominations and sects that disagree on topics ranging from the true divinity of Jesus, the immaculate conception, the resurrection and various other issues of scripture (literal versus figurative) and politics/morality (such as abortion, homosexuality, polygamy etc).
The majority of Christian doctrine sees Jesus as "son of God", that is, a sort of incarnation of the Judaic God who lived during the time of around 0 CE to 30 CE (though these exact years are doubly complicated by being defined based on the Birth of Jesus and large errors in the calendar). More progressive branches of Christianity do not necessarily see Jesus as being an actual son of God; much like Islam (or some sects of Judaism) Jesus is instead seen as a teacher or prophet of God. Regardless, Christianity began between the birth of Jesus Christ, or sometime after his death/resurrection, depending on who you ask.
Note: Like any history mixed with religion, there exists very different interpretations from sides which have much to gain by proving themselves right. Both sides of any argument will probably have "historic" evidence that proves them right, so it's mostly up to which author you believe, or which interpretation of the events makes more sense to you. Whenever possible, attempts have been made to discuss both the church's take on things and various suggested alternatives.
The clearest influence comes from Judaism, given that Jesus Christ was raised as such and was considered a Rabbi, most likely of the Essene sect. Given the current state of Roman occupation at the time, Hellenism (a combination of the Roman pantheon and the Greek pantheon) also existed as a clear outside influence.
A more controversial (and oft-disputed) link exists between Christianity and the worship of Mithras, a god in the pantheon of Zoroastrianism. Parallels have been made between aspects of modern Christianity such as December 25th and communion and rites involving Mithras. Even the tale of the "wise men" (Magi) who visit the infant Jesus has connections to Zoroastrianism, since the Magi were traditionally of the Priestly caste of Zoroastrianism.
Jesus the Biblical figure
Jesus the historical figure
Constantine and Nicaea
Constantine was emperor of Rome from 306 to 337 CE. Constantine's mother was a member of one of the various Christian sects at the time, so Constantine was considerably more lenient on their existence.
Traditionally, Christians regard Constantine as the emperor who converted to Christianity; he often used the cross as his symbol, and dedicated his victory to Jesus (Jesu).
Given Constantine's hellenistic beliefs, however, it would have been easy for him to see the parallels between Jesus Christ and Mithras, and since Mithras was already associated with Apollo and Helios (all members of Sol Invictus), it made sense to include Jesus as yet another representative. Thus, Jesus simply became incorporated into the Hellenistic pantheon at the time, with tributes made like any other. This explains the presence of Jesus among Hellenistic gods in some artwork, and has led some to theorize that the modern icon of Jesus (who is clearly not Semetic in appearance) is actually a representation of Apollo.
The Middle Ages
Luther and Protestantism
Since Luther split from the church, Christianity has been divided into Roman Catholicism, the various "Orthodox" churches (Russian, Greek, Coptic etc.) and Protestantism. The Catholic Chuch has mostly remained in one piece, though there are a few breakaway factions who claim connection (despite the denials of the Vatican), similar to the various Orthodox churches. Meanwhile, there are countless sects (or denominations) based upon Protestantism.
The "original" breakaway sect would be the Lutheran church. Other large mainstream denominations (which generally agree with each other more often than not) include the Presbyterian, Methodist and Anglican churches; these churches have generally taken a more liberal approach (to varying degrees) than the Catholic church. Along the more conservative lines lie groups such as the Baptists, the Evangelical movement, the Pentecostals, the Bretheren, the Jehova's Witnesses, and The Seventh Day Adventists. Also in a category all its own are the Mormons, who have significantly deviated from most branches of Christiantiy while maintaining its core concept.
Besides that, there are numerous smaller groups that dot the landscape; some of these groups are labelled as cults while others are simply small enough the fly under the radar of the public eye. Such groups often form due to differences between a part of a congregation and a major church.
Christianity and Spirituality
Almost all sects of Christianity are monotheistic in nature; that is, God, Jesus Christ and the Holy Spirit (together referred to as the holy trinity) make up a single deity. A curious issue arises concerning the existence of other gods; namely, contemporary Christian doctrine does not believe in other gods, but there are various biblical references to their existence. Mostly, however, these issues are not addressed, or the other gods are identified as "false gods".
The existence and source of magic is an issue that results in conflicting ideas within Christianity. Many adherents do no believe in magic per se, but Christianity does incorporate miracles in its contemporary doctrine. Also, acts of magic are mentioned within the bible, and such acts are forbidden, which begs the question of how Christianity cannot accept the existence of magic, yet still describes it and forbids it.
Generally speaking, mainstream Christianity allows for supernatural occurrences in the form of miracles; that is, acts of God performed either independently or through someone. Many conservative groups (Baptist and Roman Catholic variants in particular) believe that individuals to this day have supernatural Gifts given to them by God. There is a fairly standard set of abilities accepted as Gifts, using apostalic miracles from the bible as examples.
As for acts of magic performed by others, opinions vary; liberal factions may consider it some sort of tapping of a universal divine source, midline factions will write it off as ineffective, and conservative groups can consider it to be Satan acting in the guise of whoever or whatever the mage is evoking/invoking.
Prayer is meant to be a form of direct communication with the divine, generally intended to be God. Though prayer can be directed at other figures in Christianity (most commonly saints), proper doctrine states that one is not directly praying to the saint, but having said figure be a go-between of sorts; this is because prayer is not only meant to be communication, but a form of worship as well, and worshiping a figure other than God in traditional Christianity is forbidden. Such prayers are often a combination of thanks and requests.
In terms of origins, prayer harkens back to the Judaic precursor faiths as offerings to the divine, continuing through Judaism and into Christianity. Such prayers have always been a part of religious ritual, either with the involvement of some special person (priest, rabbi, etc.) or by the individual.
In Christianity, Prayer is allowed to be a very informal act, though "The Lord's Prayer" has become the default for many. It should be noted, in a touch of irony, that said prayer was meant to be an example of doing your own thing, but somehow the prayer became doctrine. Regardless, prayer in Christianity is significant because it allows for a day-to-day personal communication with God.
Though Saints are technically speaking a matter of the Catholic church, their role in all of Christianity is widespread enough to mention here.
In Catholic doctrine, saints are people that through their actions in life have shown themselves to be of exceptional devotion or power in their involvement with the church. These people are believed to have continuing power on the living world after their death through miracles. In a way, God is said to have bestowed extra recognition upon these people for their actions on the earth.
Like many "exceptional" issues in Christianity, Saints often represent a delicate situation for the church, due to their "elevated" status. Saints are not worshiped or prayed to, though their images and icons are often places of prayer and reverence. Instead, they are seen as a source of inspiration or guidance; someone to pray with instead of to. Still, saints and their images often become quite powerful symbols.
Previously, saints were decided upon by public acclaim. Simply put, the origin of saints comes from remembering the people who had made a difference (or those who had been martyred for their faith), and this eventually became enshrined as Saints. Pope John Paul II changed this in 1983 by making it a three-step process. The first step was veneration, where after the person's death a petition would be made to look at their works for evidence of good deeds (or martyrdom) and orthodoxy of belief. The second step is beatification, which occurs either after one documented miracle on the person's behalf, or if the person had been martyred; now the person is considered blessed. Finally, canonization requires one more miracle after beatification regardless of if they'd been martyred or not. At this point they are declared a saint.
There are over 10,000 martyred, canonized, blessed or venerated people in the history of Christianity. The number is unclear due to a lack of old records (especially of martyrs), figures who may or may not be simply legends, or multiple saints actually being the same person under different names. The Catholic church has actively tried to remove the saints that appear to be pure fiction, a move which resulted in the removal of various saints from official celebration (most famously St. Christopher), though they may still be celebrated locally. Most saints have one or more patron associations with them, usually based on their works in life.
An excellent listing of many saints can be found at http://www.catholic-forum.com/saints/indexsnt.htm; though not official it's a pretty large listing and useful for finding patron saint associations.
Yet another source of contention amongst various sects of Christianity is the nature of the afterlife. The simple answer is that the vast majority of the Christian sects believe that some time after death the person is either rewarded by going to Heaven or punished by going to Hell. After that, however, the divisions begin.
The exact timing of such a passage to the afterlife is one issue; more conservative branches tend to interpret the rise to the afterlife as an occurrence associated with the end times (as outlined in the book of Revelations), and therefore all the dead are currently interred waiting for the rapture. Others take a more liberal view, saying that the passage occurs upon the death of the individual.
The locations of the afterlife are also a matter of opinion; besides heaven and hell, purgatory and limbo have also existed as part of Catholic belief. In Purgatory, those who were not bad enough to deserve Hell go to Purgatory where they are punished/cleansed for their sins, and will eventually get to go to Heaven. Limbo existed for those who were good but had never been given a chance to be baptized (making it a common destination for children and infants). Unlike Purgatory, there wasn't an implied eventual passage to Heaven, though upon judgement day such a passage may have occurred. Either way, Purgatory and Limbo as doctrine for the Catholic church have both been downplayed in recent times. Neither Purgatory nor Limbo have any place within the major Protestant sects.
Probably the most divisive part of Heaven/Hell is the criteria for entry; in fact, this was one of the major sources of the Protestant/Catholic split as outlined by Luther's theses. Among these divisions is the source of the salvation (church versus individual) and the criteria (deeds versus faith). In theory, Protestant doctrine generally stated that accepting Christ was the prerequisite for entry into heaven (coupled with a pious life after the fact), but more liberal doctrines widen the margins to allow for those who have done "good" things in life but have not been part of the church. Conservative groups preach that no matter how good one is, Hell awaits those who aren't Christian. Catholic doctrine requires interacting with a church/priest in order to enter heaven; mass, communion and confession are all part of this, as salvation is granted through the church. Once again, this produces friction between branches, though the more liberal branches tend to acknowledge that either should work.
Given that Jesus Christ was crucified, he is part of Tiphareth, the sphere of dead or murdered gods. As detailed above, Mithras, Apollo and Helios (members of Sol Invictus have been associated with Jesus as well.
Another figure often said to be associated with Jesus is Osiris, most notably due to both experiencing death and resurrection. There are plenty of lists of comparison between the two floating around; however, the detractors claim that many of the similarities do not exist or are based on a very liberal interpretation of texts at best, and the reports claiming the association don't provide sufficient documentation to look up the source texts. With this in mind, people make the association, whether the historical evidence backs it up or not.
It should also be noted that one can find theories associating Jesus and Isis, Horus, Set, Krishna, Dionysus (Bacchus), Attis and Zoroaster (not to mention a joke list connecting Jesus and Elvis), all based on "historical evidence", usually with little or no references. Generally, these comparisons are made in an effort to suggest that Jesus was a fictional construct, so maybe these theories suggest an effort to turn the tables on Christianity and their revisionist histories.
On a more positive side, such attempts regardless suggest a certain universality to the story of Jesus; with such a strong connection to classical elements of myth-making and storytelling, one can make the argument it would still be a very strong form of narrative magic, whether the existence of Jesus is true or not.
There are Christian curses. For an example, see Reiver curse.
Fragments of curses invoking the names of the angels and disciples have been found. The harshest of which was "curse of a mother against her sons female companion".