Oddly, to get a scriptural answer to what Amen means, we are best to begin at a place where it is used as the first word in a sentence, not the last word...
Jesus would often say "Verily" or "Truly" to begin a statement.
In John's gospel (example John 3:3) Jesus is recorded as using the word twice in succession, "Verily, verily, I say to you...". This is actually the word Amen.
Compare an instance in Mark, with the same in Luke. Mark has left the word untranslated just as amen. But Luke has translated it as truly (Mark 9:1, Luke 9:27).
This shows us that Amen means truth and verity. It's a solemn affirmation.
When we say Amen we mean, "Yes before God I agree with that, I believe that to be true, I want that to be so".
Some other interesting and helpful passages...
Instead of saying, "Amen, Amen" Jeremiah paraphrased the second Amen: "Amen, the Lord do so" (Jeremiah 28:6).
God is called "The God of Truth". If we left the last word untranslated, the title would read, "The God of Amen" (Isaiah 65:16).
Paul uses the word as an affirmation: "For as many as may be the promises of God, in him they are yes, and in him Amen" (2Corinjthians 1:20).
Jesus is called "The Amen, the Faithful and True Witness" (Revelation 3:14).
So Amen is a solemn affirmation of truth, a special word for when we have said or heard something that is true before God.
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