to see: to take something in through the eyes, and so to notice or realize it
This is the ordinary New Testament word for seeing something happen and grasping what it means. In practice it almost always appears in its past-tense form ("saw"), and it is the verb the Gospels reach for again and again: someone sees a crowd, sees a sick person, sees Jesus, and acts on what they have seen.
The word travels naturally from the eye to the mind. To see a thing is often, in the same breath, to take in what it signifies, so "he saw" frequently shades into "he perceived" or "he realized." When Jesus "saw" people's thoughts or someone "saw" that an answer was wise, nothing physical is in view at all; the word has moved fully into the sense of understanding a situation.
It also covers seeing in the sense of going to look, visiting, or witnessing an event, and in the book of Revelation it is the steady drumbeat that introduces each new vision: "I saw." One short command form of it, often printed as "Behold!" or "Look!", is really just "see this" used to point.