to look ahead to something good with expectation, ranging from ordinary anticipation to settled trust in God
This verb points the mind forward. At its lightest it simply means to expect or count on something happening, the way a traveler counts on reaching a city or a writer counts on visiting friends soon. At its fullest it means to fix one's confidence on a person, above all on God, so that the expectation is really a form of reliance: the one who hopes is leaning on someone to come through.
The two shades share one core, a forward look toward a good outcome, which is why the same word carries both. What separates them is not the direction of the look but where the weight rests. Ordinary hope rests on circumstances and can be disappointed, as when the disciples on the road to Emmaus said they had hoped Jesus would redeem Israel, and then he was crucified. Confident hope rests on God, and the New Testament treats that kind as steady precisely because its object does not fail.
Greek marks the stronger sense with a small tell: when the word means trust, it is usually followed by a phrase meaning to hope on or hope in a person. That wording, hoping on the living God, is the New Testament's settled language for faith that looks to the future.