Oneness believers believe that for water baptism to be valid, one must be baptized in the name of Jesus, rather than the mainstream baptismal formula in the name of the Father, and of the Son, and of the Holy Spirit. They believe this follows the example found in Acts 2:38;
Then Peter said unto them, Repent, and be baptized every one of you in
the name of Jesus Christ for the remission of sins, and ye shall
receive the gift of the Holy Ghost.
"Jesus-Name" is a description used to refer to Oneness Pentecostals and their baptismal beliefs. The Oneness revival movement was sparked by a sermon given by R. E. McAlister in 1913 in which he stated,
"Apostolic baptism was administered as a single immersion in a single
name, Jesus Christ...The words Father, Son, and Holy Ghost were never
used in Christian baptism."
Churches that accept the oneness belief include: United Pentecostal Church International, Affirming Pentecostal Church International, Apostolic Assemblies of Christ, Apostolic Assembly of the Faith in Christ Jesus, Apostolic Gospel Church of Jesus Christ, Apostolic Overcoming Holy Church of God, and Assemblies of the Lord Jesus Christ.
Those who hold to the Trinitarian formula most certainly would not consider such a baptism to be valid because Trinitarians recognize Oneness believers as Modalists, and modalism was named a heresy in the third century. Considering that up until the year 1826 the Catholic Church used to execute heretics, I would assume that they would not consider those individuals Christian today unless they were re-baptized using the traditional formula.