Abba is the English transliteration of an Aramaic word (i.e. written in English as it sounds in Aramaic) which essentially means father, but in the original language expresses a much deeper affection and child-like trust in a parent. It's found three times in the New Testament, each time followed by its Greek equivalent, the two words together pronounced awb-bah, pay-ter, but translated in most English-language Bibles as "Abba, Father."
Jesus Christ used the term "Abba, Father" (actually, "Abba, Payter") when He prayed to God on the night before He was crucified on the stake (see Crossing The T):
"And they came to a place which was named Gethsemane: and He saith to His disciples, Sit ye here, while I shall pray. And he taketh with Him Peter and James and John [see also The Inner Circle], and began to be sore amazed, and to be very heavy; And saith unto them, My soul is exceeding sorrowful unto death: tarry ye here, and watch."
"And He went forward a little, and fell on the ground, and prayed that, if it were possible, the hour might pass from Him. And He said, Abba, Father, all things are possible unto thee; take away this cup from Me: nevertheless not what I will, but what Thou wilt."
And He cometh, and findeth them sleeping, and saith unto Peter, Simon, sleepest thou? Couldest not thou watch one hour? Watch ye and pray, lest ye enter into temptation. The spirit truly is ready, but the flesh is weak."
"And again He went away, and prayed, and spake the same words. And when He returned, He found them asleep again, for their eyes were heavy, neither wist they what to answer Him."
"And he cometh the third time, and saith unto them, Sleep on now, and take your rest: it is enough, the hour is come; behold, the Son of man is betrayed into the hands of sinners. Rise up, let us go; lo, he that betrayeth Me is at hand."
"And immediately, while He yet spake, cometh Judas, one of the twelve, and with him a great multitude with swords and staves, from the chief priests and the scribes and the elders." (Mark 14:32-43 KJV)
The Holy Spirit of God produces children for and of God:
"For as many as are led by the Spirit of God, they are the sons of God [see also One Is The Loneliest Number]. For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father [i.e. "Abba, Payter" as it was actually recorded]. The Spirit itself beareth witness with our spirit, that we are the children of God: And if children, then heirs; heirs of God [see also Raising Children], and joint-heirs with Christ; if so be that we suffer with Him, that we may be also glorified together" (Romans 8:14-17 KJV)
Fact Finder: How are the Ten Commandments the basis for determining that God truly is our Father?
See Seventh Commandment