Abba refers to one’s male parent, his father.

[DavidCox]

This is a term or general word signifying a relationship. This relationship can be crassly defined as the biological male parent (“father”), but the relationship as used in normal speech, and especially in the Bible refers to a relationship that accompanies and defines the “father” concept. These complex multimeanings behind the word “father” include: (1) creator or maker, as in the biological cause of the son’s existence. (2) provider and sustainer, meaning the person which takes care of the needs and cares of the son. These two meanings have to do more with obligations, duties, and responsibilities of the father figure. (3) dictator and source for moral codes and conduct models. This is essentially the person who has both the authority and wisdom to impose the lifestyle, likes, pleasures, purpose, end, etc. of the son. In this meaning of “Abba-Father”, we find it used with God the Father imposing “His will” on the Son so that the Son learns obedience. (4) The concept of “father” most strongly conveys one of an affectionate relationship between two people, usually the older being the father, and the younger being the son. Summary: The principle concept of “Abba” is one of authority, power-force-imposition, and love. The love factor accounts for much of what the father imposes. It is because of his love for the son that he works and forcefully imposing his will on his son. This presumes of course that the father both has power and wisdom to begin with.

The aramaic expression “abba” was often used in NT times (Koine Greek) as a term of affection, similar to our “dad, daddy, papa,” etc. (Mark 14:36; Rom 8:15; Gal 4.6). It is to be noted that Jesus apparently used this term frequently in his prayers with his heavenly Father.

This word is used in a combining form (“ab”) to make other names, Abimeleck, Abner, Eliab,

[ISBE] ab´a (ἀββᾶ,אבּא , ‘abbā), Hebraic-Chaldaic, “Father”): In Jewish and old-Christian prayers, a name by which God was addressed, then in oriental churches a title of bishops and patriarchs. So Jesus addresses God in prayer (Mat 11:25, Mat 11:26; Mat 26:39, Mat 26:42; Luk 10:21; Luk 22:42; Luk 23:34; John 11:41; John 12:27; John 17:24, John 17:25). In Mark 14:36; Rom 8:15, and Gal 4:6 ὁ πατήρ, ho patḗr, is appended even in direct address, in an emphatic sense. Servants were not permitted to use the appellation in addressing the head of the house. See Delitzsch on Rom 8:15; compare G. Dalman, Gram. des jüd.-palast. Aramaisch, etc., section 40, c. 3.

[Concise Bible Dictionary] The Greek form is αββα, father: it is the same as Ab in Hebrew, but was pronounced Abba in the time of our Saviour. It occurs three times in the New Testament, and is always followed by ‘father,’ and translated Abba Father; that is, the ‘abba’ is transcribed and not translated: if it were translated it would be ‘Father Father.’ In the Greek it stands thus: αββα ο πατηρ, :the ‘Abba’ being Aramaic, and the ‘Father’ Greek. In the Old Testament Ab was not restricted in its use to children. Elisha used it toward Elijah; servants applied it to their masters, &c.: see 2Ki 2:12 5:13 6:21. Jehovah asked, “Hath the rain a father?” Job 38:28. In the N. T. it appears to be used in a stricter sense of relationship: “Ye have received the Spirit of adoption or sonship, whereby we cry, Abba Father,” Rom 8:15 ; and “because ye are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba Father.” Ga 4:6. The only other instance is when the Lord thus addresses His Father, Mark 14:36 ; and the Spirit in the hearts of believers puts the very words He used into their lips. It has been suggested that in the two words the Jew and the Gentile each say ‘Father’ in his own language-the Aramaic being then spoken by the Jews, and Greek the language of the Gentiles in Palestine and many other places. God had been revealed in the Old Testament as Jehovah, the Almighty, &c., but it was reserved for New Testament times for Him to be made known to believers in the relationship of Father: cf. John 20:17.

[Bridgeway] Abba was a common word in the Aramaic and Hebrew languages, and meant ‘father’. It was a warm and informal term used in the everyday language of family life. Jews of Old Testament times never used abba when addressing God, but Jesus used it when praying to his Father (Mar 14:36). The early Christians also addressed God as Abba; for, through Christ, God has adopted believers as his sons and made them joint heirs with Christ of his heavenly inheritance (Rom 8:15-17; Gal 4:5-6; cf. 3:26; see ADOPTION).

[Faussett] The Chaldaic-Hebrew form, as ab is the Hebrew form, for the Greek pater, “father.” Instead of the definite article which the Hebrew uses before the word, the Chaldee or Aramaic adds a syllable to the end, producing thus the emphatic or definitive form. It is used to express a vocative case, and therefore is found in all the passages in which it occurs in the New Testament (being in all, an invocation): Mar 14:36; Rom 8:15; Gal 4:6.

The use of the Hebrew and of the Greek appellation addressed to the one Father beautifully suggests that the Spirit of adoption from Jesus, who first used the double invocation, inspires in both Jew and Gentile alike the experimental knowledge of God as our Father, because He is Father of Jesus with whom faith makes us one, and as our God because He is Jesus’ God. Compare Joh 20:17, “ascend unto My Father and (therefore) your Father. and to My God and (therefore) your God”; Gal 3:28, “there is neither Jew nor Greek, for ye are all one in Jesus Christ”; Eph 2:18, “through Him we both have access by one Spirit unto the leather.” (Especially (See ABADDON above.) “Abba” was a title not to be used by slaves to a master, nor Imma to a mistress, only by children: see Isa 8:4. “Before the child shall have knowledge to cry Abi, Immi.”

[Easton] This Syriac or Chaldee word is found three times in the New Testament (Mark 14:36; Rom. 8:15; Gal. 4:6), and in each case is followed by its Greek equivalent, which is translated “father.” It is a term expressing warm affection and filial confidence. It has no perfect equivalent in our language. It has passed into European languages as an ecclesiastical term, “abbot.”

[Amtrac] When the Jews came to speak Greek, this word may have been retained from their ancient language, as being easier to pronounce, especially for children, than the Greek pater

Abba
David Cox’s Topical Bible Concordance
• Father. Mar. 14:36; Rom. 8:15; Gal. 4:6.