Roman Catholic views
Confirmation is seen as granting the receiver an extra-natural source of wisdom, knowledge and courage, should the person desire it with an open heart. As such, Confirmation is the fulfillment of the words of Christ who said "And ye shall know the truth". See also the New Testament Gospel of Saint John, chapter 14 where Christ discusses the topic with the Apostles.
As a sacrament it is held to put a person in direct communion with the Holy Spirit. The current 'witness' paradigm of Confirmation used in the Latin Rite since the seventies emphasizes participation in the community and the evangelistic nature of the grace obtained. Prior to that the 'soldier of Christ' paradigm emphasized defending Christianity and so was more concerned with the apologetic nature of the graces.
In Latin-Rite (i.e., Western) Catholic churches, its usual minister is a diocesan bishop; a confirmation by a parish priest would usually be illegal, but nonetheless valid (unlike a confirmation by an unordained person, which would be both illegal and invalid). In Western Catholic churches, only persons old enough to understand the sacrament are normally confirmed. In Eastern-Rite Roman Catholic churches, the usual minister of this sacrament is a parish priest, who uses the chrismation rite involving olive oil consecrated by a bishop (i.e., chrism), and administers the sacrament immediately after baptism.
Its supernatural efficacy is held to depend on its being administered by a person ordained by a bishop in a line of succession of bishops dating back to the twelve apostles. Roman Catholics recognize the validity of chrismations in Eastern Orthodox and Oriental Orthodox churches because those churches maintain the apostolic succession of bishops, and therefore Catholics do not confirm converts from those churches who have been chrismated before their conversion.
Orthodox views In Eastern Orthodox, and Oriental Orthodox churches, there is no direct theological analogue to "confirmation". The Orthodox churches practice Chrismation as part of the normal baptism, since it is seen as part-and-parcel, quite inseparable, in the Orthodox theology of baptism. When Roman Catholics (and some Protestants) convert to Orthodoxy, they are admitted via Chrismation alone, but this is a matter of local Episcopal discretion, and a bishop may require all converts be admitted by baptism if he deems it necessary. (Depending upon the form of the original baptism, some Protestants must be baptized upon conversion to Orthodoxy.)
In Protestant churches, confirmation is often called a "rite" rather than a sacrament, and is held to be merely symbolic rather than an effective means of conferring divine grace. Catholics do not recognize the validity of Protestant confirmations, and therefore do confirm converts from Protestantism.
Each person is confirmed only once
Western Christians do not normally confirm anyone who has already been confirmed, just as they do not typically baptize anyone twice. Roman Catholics have made it an explicit dogma that confirmation is one of the three sacraments that no one may receive more than once; see sacramental character. On the other hand, the Orthodox insist that chrismation upon conversion is necessary.