father, spoken as a direct and trusting address: the Aramaic word for one's father, used to call on God as Father
"Abba" is not a Greek word at all but an Aramaic one, spelled out in Greek letters. It is the everyday Aramaic word for one's father, and it turns up in the New Testament only three times, always in the same fixed pair: the Aramaic word followed at once by its Greek twin, πατήρ, "the Father." Jesus prays it in Gethsemane, and Paul twice quotes it as the cry of Christians at prayer.
In the Aramaic of Jesus' day this was simply how a person said "Father!" out loud. Grown sons and daughters used it of their own fathers, not only small children, and it was respectful enough to serve as a title for honored teachers. So the word is warm and familial, the language of the family table rather than the courtroom, but it is not baby talk.
The reason the early church held onto the Aramaic word, instead of simply translating it, is that it was Jesus' own word. His followers prayed the way he had taught them, keeping the very sound he used and setting the Greek πατήρ beside it. That two-language phrase became a shared prayer, and it carries a startling claim: that ordinary believers may speak to God with the same trust the Son did.