The definition below is taken from Vine’s Expository Dictionary Of Old & New Testament Words. An excellent reference when studying the Holy Bible. (Another very good [electronic] reference book is The Strong’s Concordance.)

ABBA

ABBA is an Aramaic word, found in Mark 14:36; Romans 8:15 and Galatians 4:6. In the Gemara (a Rabbinical commentary on the Mishna, the traditional teaching of the Jews) it is stated that slaves were forbidden to address the head of the family by this title.  It approximates to a personal name, in contrast to “Father,” with which it is always joined in the N.T.  This is probably due to the fact that, “Abba” having practically become a proper name, Greek-speaking Jews added the Greek work pater, father, from the language they used.  “Abba” is the word framed by the lips of infants, and betokens unreasoning trust; “father” expresses an intelligent apprehension of the relationship.  The two together express the love and intelligent confidence of the child.¹

¹Vine, W.E. (1997). Vine’s Expository Dictionary Of Old & New Testament Words. Nashville, TN: Thomas Nelson, Inc.