In outline, the New Testament understands God to have three "faces": God the Father, who is the creator of heaven and earth; Jesus, his son, who is God made manifest or real as a human being; and the Holy Spirit, which is the invisible, guiding spirit of God that people in the world today have access to, especially Christian believers.
These three aspects of the God are equal: this means that no one aspect of God is supreme or rules over the other. The three are often described in theology as in a state of perichoresis, or an intimate dance, with each other.
Trinitarian theology has its roots in the Bible but was developed over time by the Church fathers, who needed to reconcile the idea that God, Jesus, and the Holy Spirit were one in order to keep Christianity monotheistic. The Gospel of John claims that Jesus existed, like God, before the creation of the world. The New Testament, and notably, again the Gospel of John, claims that the Holy Spirit is the ever-present, living, spirit of God. It was important to the early Church that Jesus and the Holy Spirit be understood not as God's subordinates but as his equal partners.
What trinitarianism does is explain that Christianity, like Judaism, is monotheistic, and that though Christians worship God the father, Jesus, and the Holy Spirit, they are worshipping one God, not three. The trinitarian doctrine embedded in the Nicene Creed, a foundational Christian doctrine, insists that the three are only different aspects of the same God. Just as photos of you from the back, from the front, and in profile are all of the same person, so too are God, Jesus, and the Holy Spirit different aspects of one God. Christian theology has said that it doesn't make sense to say that worshipping Jesus or the Holy Spirit is different from worshipping God the Father.