To start, it is instructive to understand that the words "faith", "trust" and "believe" in English translations all come from the same Greek root (pisteuo).
Charles Spurgeon describes it well in his book, All of Grace:
What is faith? It is made up of three things—knowledge, belief, and trust.
Knowledge comes first. "How shall they believe in him of whom they have not heard? I want to be informed of a fact before I can possibly believe it..."
The mind goes on to believe that these things are true. The soul believes that God is, and that He hears the cries of sincere hearts; that the gospel is from God; that justification by faith is the grand truth which God hath revealed in these last days by His Spirit more clearly than before. Then the heart believes that Jesus is verily and in truth our God and Saviour, the Redeemer of men, the Prophet, Priest, and King of His people. All this is accepted as sure truth, not to be called in question...
So far you have made an advance toward faith; only one more ingredient is needed to complete it, which is trust. Commit yourself to the merciful God; rest your hope on the gracious gospel; trust your soul on the dying and living Saviour; wash away your sins in the atoning blood; accept His perfect righteousness, and all is well. Trust is the lifeblood of faith; there is no saving faith without it.
He later summarizes it this way:
Faith is believing that Christ is what He is said to be, and that He will do what He has promised to do, and then to expect this of Him.
In reality, the Greek word is not anything mysterious. However, it does not mean intellectual assent, as the demons believe there is a God--and tremble in fear as a result.
So, the definition comes from understanding the Gospel--what is to be believed and why.