What are some specific verses that explain the Trinity?

There are no verses in the Bible that explain the Trinity. (The Trinity defies human explanation.) However, there are many passages in God's Word that reveal the Trinity.

One God

It is clear from the Bible that there is only one God. That is clear enough from passages such as Deuteronomy 6:4, Isaiah 43:10 and 1 Timothy 2:5 (among others). There are not 2 gods, or 3 or more. There is just one God.

Three Persons

The next thing that is clearly revealed in Scripture is that there are three Persons, who are all called God and shown to have attributes that could only be possessed by God. Those three Persons are revealed as God the Father (John 6:27), Jesus Christ the Son (Colossians 1:16-17) and the Holy Spirit (John 14:26).

Some Bible passages refer to all three Person in the same thought, such as the following:

...according to the foreknowledge of God the Father, in the sanctification of the Spirit, for obedience to Jesus Christ and for sprinkling with his blood: May grace and peace be multiplied to you. 1 Peter 1:2 (ESV)

Notice Peter speaks of the Father, the Spirit and the Son (Jesus Christ) all in one thought. This can also be seen in the following passage:

The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all. 2 Corinthians 13:14 (ESV)

The doctrine of the Trinity reveals that the Father is God, the Son is God, and the Holy Spirit is God—and yet there is only one God. That is the essence of what the Bible reveals concerning the nature of God. And yet belief in the Trinity is a huge stretch for a lot of people — mostly owing to the fact that it extends beyond the range of our human comprehension. There are a lot of people who struggle to embrace anything they can't grasp on an intellectual level. That's unfortunate since they are essentially demanding that God's nature must be understandable in order to be believed.

A very wise Bible teacher once told me whenever I'm dealing with God to always leave room for mystery. I have found that to be incredibly insightful advice.