If you would like the to know the history of the concept of the trinity (God being 3 in 1) and the stances taken on it over time, as well as the verses in the Bible that point to God both being one and yet also composed of Jesus, the Holy Spirit, and the Father, Wikipedia has a great answer, just look up the word “Trinity”. As far as the traditional understanding of the Trinity, it is the belief that God in the persons of the Father, Son (Jesus) and Holy Spirit are one, were never created, have always existed, and have always been ONE in nature or essence. Jesus is believed to have existed before he took human form and was present during creation. Multiple bible verses are pointed to for the defense of this concept. I will just give a a couple here in a very cursory explanation.
One of the main verses pointed to is Jesus saying, “I tell you the truth, before Abraham was born, I Am.” John 8:58. Not only is this a claim of the pre-incarnate (before he became man) existence of Jesus, but also a claim to his deity when you remember that God told Abraham his name was “I AM.”
Jesus also said, “If you really know me, you will know my Father as well. From now on, you do know him and have seen him.” John 14:7
In Genesis 1:26 God refers to himself in the plural, and the word for God in Hebrew is Elohim, which is the plural for God. A caveat to this is that in Hebrew it is not uncommon to make words plural for emphasis of importance or grandeur, which is why historically this has not bothered the Jewish understanding of God being one being despite this verse. One of the central tenants of Judaism is that there is only one God. However, Christians tend to take the plural in this verse as early evidence of the Trinity.
There is so much more, including more verses in the gospels and the Pauline epistles, as well as early church fathers that mention this concept as well.
The issue is that the Bible and the early church seemed to affirm that God was one and yet that Jesus and the Holy Spirit were also God, along with the Father. How these two concepts fit together was something the early church tried to figure out and fought over. Its something theologians say is still beyond our grasp and, while there are many ways we try to explain it through diagrams or objects such as the Shamrock or an egg, we fall short of ever perfectly representing the nature of the Trinity.
Hope that helps.