There is one God in three persons: Father, Son, and Holy Spirit. That's the trinity in a nutshell. I knew what the trinity was from a very young age, though I'm not sure exactly how I knew it. I remember singing the doxology at church every week: "Praise Father, Son, and Holy Ghost." However, I can't recall ever being taught about the trinity at church in any format. Maybe I was and I just don't recall it, but it certainly was not talked about often.
In the second semester of my junior year in college, I took a class on the doctrine of God. One of our assignments was to rate the importance of the doctrine of the trinity on a scale of 1 to 10 and defend our rating. If I remember correctly, I rated it a 6. I had two primary reasons for that rating. 1) The Trinity is nowhere explicitly taught in the Bible. You won't find the word "trinity" in your Bible, nor language of "persons" or God being "three in one." 2) I had not ever (at that point in time) brought up the trinity while witnessing to a lost person. It might be important, but not as important as Christ's atoning death, or the resurrection of the dead, or salvation by grace through faith in Christ, etc, etc...
That class discussion did not go well for me. In fact, the next 3 or 4 classes did not go well for me. However, it did show me a blindspot and forced me to investigate my beliefs. My mind didn't change quickly, but it did change. Now, I'd say that I think the trinity is among the very most important doctrines of our faith, and I decided to take several weeks to try to explain it to our student ministry. Here's why I came to that conclusion.
If you study the history of the church, you will quickly find that the trinity was a source of heated debate for the first four or five centuries. It was hotly debated because the identity of God was at stake. The Scriptures had clearly taught that there was one God (Deut. 6:4), but what about Jesus? Jesus did and said things that only God should do or say, so was there a second god? Perhaps a lesser, created god? What the church repeatedly concluded was that God should be thought of as a trinity: one substance with three persons (read the Nicene Creed and the Athanasian Creed to see what they taught in their own words). Now, the church fathers were just men, and men get things wrong. They were certainly wrong about some things and did not agree on everything, just like church people still can't agree on everything. However, since after centuries of debate, this is what almost everyone agreed on, which probably means that they got what the Bible was trying to say.
I think the trinity is sort of like the glue that holds the Bible together. It's the only way to make sense of everything the Bible has to say about God. If you take the trinity away, how do you make sense of the claim that there is one God, but that Jesus is also God, and that the Spirit is mentioned right alongside the Father and the Son? This isn't the place to dive into all of the details, but the Bible turns into a bit of a mess without the trinity to make sense of everything it says about God and Jesus.
Besides making sense of the Bible, the Trinity also makes sense of the gospel. The trinity allows us to say that the baby who was born to Mary in Bethlehem was actually God Himself in the flesh. Jesus was not a powerful angel or demigod or any other created thing sent to do God's bidding. If that were the case, the cross would be nothing more than a horrible act of abuse towards one of God's creatures. If, however, Jesus was God in the flesh, then we don't have an abuse victim, but rather a willing sacrifice. God Himself was willingly suffering the consequences of sin on the behalf of sinners. He lived the perfect, substitutionary life as a human to heal and restore humanity to what God intended us to be. That is good news worth sharing!