Triunity: Some Penticostal Holiness congregations hold to the Oneness Doctrine which is virtually the same as many Messianic Jewish Congregations.
Trinity: The Other view of the Trinity is more like fundamental Christianity holding to the three persons godhead.
We at Zion believe in Abba b’shamayim (The Father in heaven), and we believe in Yeshua as Messiah son of Joseph (Talmud, Succa 52a) in whom G-d lived., and we know that G-d is Ruach (spirit) who comes to dwell in each of us if we are willing. But we do not see Him as three separate individuals. Because G-d is the Almighty, there is nothing impossible to Him, He can manifest in as many different forms as He desires and even do it simultaneously should he choose to do so. This does not mean that He is a plurality. We also see Him manifesting as a burning bush on the mountain, covering cloud by day and burning pillar of flame at night for Israelis leaving Egypt, and even manifesting as a man before Moshe and Joshua. This does not make Him Seven individuals does it? Or maybe it does! (Revelations 4:6 seven spirits of G-d) We think you should decide this for yourself.
For an excellent article on the subject of Trinity/Triunity click here