For a sacrament to be valid, three things have to be present: the correct form, the correct matter, and the correct intention. With baptism, the correct intention is to do what the Church does, the correct matter is water, and the correct form is the baptizing "in the name of the Father and of the Son and of the Holy Spirit" (Matt. 28:19).
Unfortunately, not all religious organizations use this form. In fact, Jehovah’s Witnesses sometimes use no formula at all in their baptisms, and an even larger group, the "Jesus Only" Pentecostals, baptize "in the name of Jesus." As a result, the baptisms of these groups are invalid; thus, they are not Christian, but pseudo-Christian.
Both groups also reject the Trinity. Jehovah’s Witnesses claim that Jesus is not God, a heresy known as Arianism (after its fourth-century founder), and the "Jesus Only" Pentecostals claim that there is only a single person, Jesus, in the Godhead, a heresy known as Sabellianism (after its inventor in the third century; see the Catholic Answers tract, God in Three Persons).
"Jesus Only" Pentecostals note that Jesus told the apostles to baptize in "the name" (singular) of the Father, the Son, and the Spirit, but they make the mistake of assuming that name is Jesus. There may not be a single name that Jesus has in mind at all, just as when we say, “Stop! In the name of the law,” we do not have a personal name in mind. If he did have such a name in mind, it may have been something such as God or Yahweh or Lord.
"Jesus Only" Pentecostals also argue that the New Testament talks about people being baptized "in the name of Jesus," but there are only four such passages (Acts 2:38, 8:16, 10:48, and 19:5). Further, these passages do not use the same designation in each place (some say "Lord Jesus," other say "Jesus Christ"), indicating that they were not technical formulas used in the baptism but simply descriptions by Luke. These four descriptions are not to be considered as a substitute for or contradiction of the divine command of the Lord Jesus Christ to: "make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit" (Matt. 28:19).
Rather, the phrase "baptized in the name of Jesus" is simply Luke’s way to distinguish Christian baptism from other baptisms of the period, such as John’s baptism (which Luke mentions in Acts 1:5, 22, 10:37, 11:16, 13:24, 18:25, 19:4), Jewish proselyte baptism, and the baptisms of pagan cults (such as Mithraism). It also indicates the person into whose Mystical Body baptism incorporates us (Rom. 6:3).
The early Church Fathers, of course, agreed. As the following quotes illustrate, Christians have from the beginning recognized that the correct form of baptism requires one to baptize "in the name of the Father and of the Son and of the Holy Spirit."
"After the foregoing instructions, baptize in the name of the Father, and of the Son, and of the Holy Spirit, in living [running] water. If you have no living water, then baptize in other water, and if you are not able in cold, then in warm. If you have neither, pour water three times on the head, in the name of the Father, and of the Son, and of the Holy Spirit. Before baptism, let the one baptizing and the one to be baptized fast, as also any others who are able. Command the one who is to be baptized to fast beforehand for one or two days" (Didache 7:1 [A.D. 70]).
Tatian the Syrian
"Then said Jesus unto them, ‘I have been given all authority in heaven and earth; and as my Father has sent me, so I also send you. Go now into all the world, and preach my gospel in all the creation; and teach all the peoples, and baptize them in the name of the Father and the Son and the Holy Spirit; and teach them to keep all whatsoever I commanded you: and lo, I am with you all the days, unto the end of the world’ [Matt. 28:18-20]" (The Diatesseron 55 [A.D. 170]).
"When the one being baptized goes down into the water, the one baptizing him shall put his hand on him and speak thus: ‘Do you believe in God, the Father Almighty?’ And he that is being baptized shall say: ‘I believe.’ Then, having his hand imposed upon the head of the one to be baptized, he shall baptize him once. Then he shall say: ‘Do you believe in Christ Jesus . . . ?’ And when he says: ‘I believe,’ he is baptized again. Again shall he say: ‘Do you believe in the Holy Spirit and the holy Church and the resurrection of the flesh?’ The one being baptized then says: ‘I believe.’ And so he is baptized a third time" (The Apostolic Tradition 21 [A.D. 215]).
"After his resurrection he promises in a pledge to his disciples that he will send them the promise of his Father; and lastly, he commands them to baptize into the Father and the Son and the Holy Ghost, not into a unipersonal God. And indeed it is not once only, but three times, that we are immersed into the three persons, at each several mention of their names" (Against Praxeas 26 [A.D. 216]).
"The Lord himself told his disciples that they should baptize all peoples in the name of the Father and of the Son and of the Holy Spirit . . . for indeed, legitimate baptism is had only in the name of the Trinity" (Commentary on Romans 5:8 [A.D. 248]).
The Acts of Xantippe and Polyxena
"Then Probus . . . leapt into the water, saying, ‘Jesus Christ, Son of God, and everlasting God, let all my sins be taken away by this water.’ And Paul said, ‘We baptize thee in the name of the Father and Son and Holy Ghost.’ After this he made him to receive the Eucharist of Christ" (Acts of Xantippe and Polyxena 21 [A.D. 250]).
Cyprian of Carthage
"He [Jesus] commanded them to baptize the Gentiles in the name of the Father and of the Son and of the Holy Spirit. How then do some say that though a Gentile be baptized . . . never mind how or of whom, so long as it be done in the name of Jesus Christ, the remission of sins can follow—when Christ himself commands the nations to be baptized in the full and united Trinity?" (Letters 73:18 [A.D. 253]).
Eusebius of Caesarea
"We believe . . . each of these to be and to exist: the Father, truly Father, and the Son, truly Son, and the Holy Ghost, truly Holy Ghost, as also our Lord, sending forth his disciples for the preaching, said, ‘Go teach all nations, baptizing them in the name of the Father and of the Son, and of the Holy Ghost.’ Concerning whom we confidently affirm that so we hold, and so we think, and so we have held aforetime, and we maintain this faith unto the death, anathematizing every godless heresy" (Letter to the People of His Diocese 3 [A.D. 323]).
Cyril of Jerusalem
"You were led by the hand to the holy pool of divine baptism, as Christ was carried from the cross to this sepulcher here before us [the tomb of Jesus at Jerusalem]. And each of you was asked if he believed in the name of the Father, and of the Son, and of the Holy Spirit. And you confessed that saving confession, and descended three times into the water, and again ascended, and in this there was suggested by a symbol the three days of Christ’s burial" (Catechetical Lectures 20:4 [A.D. 350]).
"And the whole faith is summed up, and secured in this, that a Trinity should ever be preserved, as we read in the Gospel, ‘Go ye and baptize all the nations in the name of the Father and of the Son and of the Holy Ghost’ (Matt. 28:19). And entire and perfect is the number of the Trinity (On the Councils of Arminum and Seleucia 2:28 [A.D. 361]).
Basil the Great
"The Holy Spirit, too, is numbered with the Father and the Son, because he is above creation, and is ranked as we are taught by the words of the Lord in the Gospel, ‘Go and baptize in the name of the Father and of the Son and of the Holy Ghost.’ He who, on the contrary, places the Spirit before the Son, or alleges him to be older than the Father, resists the ordinance of God, and is a stranger to the sound faith, since he fails to preserve the form of doxology which he has received, but adopts some newfangled device in order to be pleasing to men" (Letters 52:4 [A.D. 367]).
Ambrose of Milan
"Moreover, Christ himself says: ‘I and the Father are one.’ ‘One,’ said he, that there be no separation of power and nature; but again, ‘We are,’ that you may recognize Father and Son, forasmuch as the perfect Father is believed to have begotten the perfect Son, and the Father and the Son are one, not by confusion of person, but by unity of nature. We say, then, that there is one God, not two or three gods" (The Faith 1:1[9–10] [A.D. 379]).
Gregory of Nazianz
"But not yet perhaps is there formed upon your soul any writing good or bad; and you want to be written upon today. . . . I will baptize you and make you a disciple in the name of the Father and of the Son and of the Holy Ghost; and these three have one common name, the Godhead. And you shall know, both by appearances and by words that you reject all ungodliness, and are united to all the Godhead" (Orations 40:45 [A.D. 380]).
"[S]eeing that a man, baptized in the name of the Father and the Son and the Holy Ghost, becomes a temple of the Lord, and that while the old abode is destroyed a new shrine is built for the Trinity, how can you say that sins can be remitted among the Arians without the coming of the Holy Ghost? How is a soul purged from its former stains which has not the Holy Ghost?" (Dialogue Against the Luciferians 6 [A.D. 382]).
Gregory of Nyssa
"And we, in receiving baptism . . . conceal ourselves in [the water] as the Savior did in the earth: and by doing this thrice we represent for ourselves that grace of the resurrection which was wrought in three days. And this we do, not receiving the sacrament in silence, but while there are spoken over us the names of the three sacred persons on whom we believed, in whom we also hope, from whom comes to us both the fact of our present and the fact of our future existence" (Sermon For the Day of Lights [A.D. 383]).
"Baptism in the name of the Father and of the Son and of the Holy Ghost has Christ for its authority, not any man, whoever he may be; and Christ is the truth, not any man" (On Baptism, Against the Donatists 4:24  [A.D. 400]).
"O Lord our God, we believe in you, the Father and the Son and the Holy Spirit. For the truth would not say, ‘Go, baptize all nations in the name of the Father and of the Son and of the Holy Spirit,’ unless you were a Trinity" (The Trinity 15:28 [A.D. 408]).
Theodoret of Cyr
"And what need is there of many words, when it is possible to refute falsehood in few? We provide that those who year by year come up for holy baptism should carefully learn the faith set forth at Nicaea by the holy and blessed Fathers; and initiating them as we have been bidden, we baptize them in the name of the Father and of the Son and of the Holy Ghost, pronouncing each name singly" (Letters 145 [A.D. 444]).
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+Robert H. Brom, Bishop of San Diego, August 10, 2004