From BiblePortal Wikipedia

Vine's Expository Dictionary of OT Words [1]

A. Verb.

Nâqam ( נָקַם , Strong'S #5358), “to avenge, take vengeance, punish.” This root and its derivatives occur 87 times in the Old Testament, most frequently in the Pentateuch, Isaiah, and Jeremiah; occasionally it occurs in the historical books and the Psalms. The root occurs also in Aramaic, Assyrian, Arabic, Ethiopic, and late Hebrew.

Lamech’s sword song is a scornful challenge to his fellows and a blatant attack on the justice of God: “… for I have slain a man to my wounding, and a young man to my hurt. If Cain shall be avenged sevenfold, truly Lamech seventy and sevenfold” (Gen. 4:23-24).

The Lord reserves vengeance as the sphere of His own action: “To me belongeth vengeance, and recompense … for he will avenge the blood of his servants, and will render vengeance to his adversaries” (Deut. 32:35, 43). The law therefore forbade personal vengeance: “Thou shalt not avenge, nor bear any grudge against the children of thy people, but thou shalt love thy neighbor as thyself: I am the Lord” (Lev. 19:18). Hence the Lord’s people commit their case to Him, as David: “The Lord judge between me and thee [Saul], and the Lord avenge me of thee: but mine hand shall not be upon thee” (1 Sam. 24:12).

The Lord uses men to take vengeance, as He said to Moses: “Avenge the children of Israel of the Midianites.… And Moses spake unto the people, saying, Arm some of yourselves unto the war, and let them go against the Midianites, and avenge the Lord of Midian” (Num. 31:2- 3). Vengeance for Israel is the Lord’s vengeance. The law stated, “And if a man smite his servant, or his maid, with a rod, and he die under his hand; he shall be surely punished” (Exod. 21:20). In Israel, this responsibility was given to the “avenger of blood” (Deut. 19:6). He was responsible to preserve the life and personal integrity of his nearest relative.

When a man was attacked because he was God’s servant, he could rightly call for vengeance on his enemies, as Samson prayed for strength, “… that I may be at once avenged of the Philistines for my two eyes” (Judg. 16:28).

In the covenant, God warned that His vengeance may fall on His own people: “And I will bring a sword upon you, that shall avenge the quarrel of my covenant …” (Lev. 26:25). Isaiah thus says of Judah: “Therefore saith the Lord, the Lord of hosts … Ah, I will ease me of mine adversaries, and avenge me of my enemies” (1:24).

B. Noun.

Nâqâm ( נָקָם , Strong'S #5359), “vengeance.” The noun is first used in the Lord’s promise to Cain: “Therefore whosoever slayeth Cain, vengeance shall be taken on him sevenfold” (Gen. 4:15).

In some instances a man may call for “vengeance” on his enemies, such as when another man has committed adultery with his wife: “For jealousy is the rage of a man: therefore he will not spare in the day of vengeance” (Prov. 6:34).

The prophets frequently speak of God’s “vengeance” on His enemies: Isa. 59:17; Mic. 5:15; Nah. 1:2. It will come at a set time: “For it is the day of the Lord’s vengeance, and the year of recompenses for the controversy of Zion” (Isa. 34:8).

Isaiah brings God’s “vengeance” and redemption together in the promise of messianic salvation: “The Spirit of the Lord God is upon me; … he hath sent me … to proclaim the acceptable year of the Lord, and the day of vengeance of our God …” (61:1-2). When Jesus announced that this was fulfilled in Himself, He stopped short of reading the last clause; but His sermon clearly anticipated that “vengeance” that would come on Israel for rejecting Him. Isaiah also said: “For the day of vengeance is in mine heart, and the year of my redeemed is come” (63:4).

King James Dictionary [2]

AVENGE, avenj'. L. vindex.

1. To take satisfaction for an injury by punishing the injuring party to vindicate by inflicting pain or evil on the wrong doer.

Shall not God avenge his own elect.  Luke 18 .

Avenge me of my adversary.

In these examples, avenge implies that the evil inflicted on the injuring party is a satisfaction or justice done to the injured, and the party vindicated is the object of the verb.

2. To take satisfaction for, by pain or punishment inflicted on the injuring party.

He will avenge the blood of his servants.  Deuteronomy 32 .

Here the thing for which satisfaction is taken is the object of the verb.

3. To revenge. To avenge and revenge, radically, are synonymous. But modern usage inclines to make a valuable distinction in the use of these words, restricting avenge to the taking of just punishment, and revenge to the infliction of pain or evil, maliciously, in an illegal manner. 4. In the passive form this verb signifies to have or receive just satisfaction, by the punishment of the offender.

Shall not my soul be avenged on such a nation as this?

 Jeremiah 5 .

Webster's Dictionary [3]

(1): (v. i.) To take vengeance.

(2): (v. t.) To take vengeance for; to exact satisfaction for by punishing the injuring party; to vindicate by inflicting pain or evil on a wrongdoer.

(3): (v. t.) To treat revengefully; to wreak vengeance on.

(4): (n.) Vengeance; revenge.