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, and translated in most English-language Bibles as "Abba, Father."
Jesus Christ used the term "Abba, Father" when He prayed to God just before His arrest on The Fateful Night:
"And they went to a place which was called Gethsemane; and He said to His disciples, "Sit here, while I pray." And He took with Him Peter and James and John [see also The Inner Circle], and began to be greatly distressed and troubled. And He said to them, "My soul is very sorrowful, even to death; remain here, and watch."
"And going a little farther, He 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 to Thee; remove this cup from Me; yet not what I will, but what Thou wilt."
"And He came and found them sleeping, and He said to Peter, "Simon, are you asleep? Could you not watch one hour? Watch and pray that you may not enter into temptation; the spirit indeed is willing, but the flesh is weak."
"And again He went away and prayed, saying the same words. And again He came and found them sleeping, for their eyes were very heavy; and they did not know what to answer Him."
"And He came the third time, and said to them, "Are you still sleeping and taking your rest? It is enough; the hour has come; the Son of man is betrayed into the hands of sinners. Rise, let us be going; see, My betrayer is at hand."
"And immediately, while He was still speaking, Judas came, one of the twelve, and with him a crowd with swords and clubs, from the chief priests and the scribes and the elders." (Mark 14:32-43 RSV)
The Holy Spirit of God produces Children Of God:
"For all who are led by the Spirit of God are sons of God. For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of sonship. When we cry, "Abba! Father!" it is the Spirit Himself bearing witness with our spirit that we are children of God, and if children, then heirs, heirs of God and fellow heirs with Christ, provided we suffer with Him in order that we may also be glorified with Him." (Romans 8:14-17 RSV)
Fact Finder: Does God send the Spirit of His Son into the hearts of Believers, so that they too pray to "Abba! Father!"?