The word is often brushed over lightly. Here is an easy-to-follow article with Hebrew and Greek spelled out for you in English.
Its basic meaning is a radical and profound change of course and direction from sin and self-rule towards God. You were going in one direction, but now you are going in the opposite direction. 180 degree turn. The whole person, body (behavior), soul and spirit, must go through this change that only God initiates and can work in you.
The Old Testament was written in Hebrew, and here are three verbs indicating repenting or turning.
1.. One verb is naḥam (pronounced nah-khahm, the ḥ is pronounced like -ch- in Bach) (used 108 times). It has two distinct meanings: “to comfort, console” and “to relent, repent, change one’s mind, be grieved.” These latter definitions can sometimes apply to God’s seeming change of mind. However, this change is never a surprise or unanticipated. Rather, it means that he has conditions attached to his promises or decrees, whether stated or implied. For example, he promises something positive, and people obey, and the promise of blessing to them is fulfilled. That is his attribute of goodness. However, if they do not obey, then his attribute of justice-judgment comes forth, and he sends a prophet to warn them. If they do not repent, he judges them. If they do repent, he relents. Therefore, God is not fickle, but humans are. He simply gauges how people behave, and then he acts accordingly.
But let’s return to the topic of repentance.
2.. Another verb is ‘anah (used 79 times), and in some contexts denotes humbling in a positive sense (repentance or contrition). Some context of ‘anah pleases God (Lev. 16:31; 23:27; Ps. 35:51).
3.. The verb shub (pronounced shoob) (used 1075 times) and basically means “to turn, return, repent, go or come back.” It mostly has a physical meaning, like turning back to a point of departure. But theologically it means the act of repentance, for example, turning from idolatry to the living God. “Return to me” (Is. 44:22).
1.. The verb metanoia (pronounced meh-tah-noi-ah) (used 34 times), and it is derived from meta– (after) and nous (mind); when the prefix meta is attached to the verb as here, it means a “change of place or condition.” In this case it denotes a radical and profound moral turn of the whole person from sin to God. Repentance goes together with the forgiveness of sins (Luke 3:3; 24:7). Sins are blotted out in repentance (Acts 3:19; 5:31; 11:18; 21:21).
The Hebrew Old Testament was translated into Greek in the third to second century B.C., which is called the Septuagint (abbreviated LXX, for seventy translators and pronounced sep-TOO-ah-gent). When the verb metanoeō (see below) appears, it usually means “change one’s mind” (Prov. 20:25; Jer. 4:28). Further, the Greek verb in the LXX that often means “to turn back or return” to God is epistrephō (pronounced eh-pee-streh-foh). Behind this concept is the covenant, and Israel departing from it or returning to it (Jer. 18:8). The stem streph– means a “turning.”
2.. In the NT, the verb metanoeō (pronounced meh-tah-no-eh-oh) is the one that the writers chose instead of epistrephō, but when they appear together, metanoeō means to turn from sin, while epitstrephō means to turn to God (Acts 3:19; 26:20).
When metanoeō is used, it means a radical turn from sin to a new way of life. “Repent of your wickedness!” says Peter to Simon the sorcerer (Acts 8:22). True repentance brings forth fruit or visible evidence showing repentance (Matt. 3:8 Acts 26:20).
Then one has to believe as one changes course. He has to believe in the gospel (Mark 1:15) or the atoning death resurrection (Acts 2:38). In fact, it his faith in Christ that enables him to repent or change course.
This turning is also called “conversion,” from the Latin noun conversio, literally meaning “turning around,” which is related to the verb converto, “to turn around, to whirl around,” especially “to turn in the opposite direction” or “to turn back” or “to direct towards.”
William Mounce summarizes metanoia and metanoeō: it is “essentially to grasp the gospel message, because it does not allow for someone to obtain salvation simply by intellectually believing that Jesus is the Son of God without repenting of sins and turning to live for him” (p. 581).
So intellectual assent or agreement or changing one’s mind is insufficient. It must be a profound moral and radical change of one’s whole life.
There is an odd teaching going around the church that there is no reference in all of Scripture that says “repent from sins” or “repented of sins” or “repenting of sins.” In reply, however, here are some verses that say it either exactly or nearly exactly. The concept is certainly biblical.
The version is the NIV (emphasis is of course added).
Job 36:10: He makes them listen to correction and commands them to repent of their evil.
Is. 59:20: “The Redeemer will come to Zion, to those in Jacob who repent of their sins,” declares the Lord.
Jer. 8:6: I have listened attentively, but they do not say what is right. None of them repent of their wickedness, saying, “What have I done?” Each pursues their own course like a horse charging into battle.
Jer. 18:8 and if that nation I warned repents of its evil, then I will relent and not inflict on it the disaster I had planned.
Ezek. 14:6 Therefore say to the people of Israel, ‘This is what the Sovereign Lord says: Repent! Turn from your idols and renounce all your detestable practices!
Luke 3:3: He went into all the country around the Jordan, preaching a baptism of repentance FOR the forgiveness of SINS.
Luke 5:32: I have not come to call the righteous, but SINNERS to repentance.”
Luke 15:7: I tell you that in the same way there will be more rejoicing in heaven over one SINNER who repents than over ninety-nine righteous persons who do not need to repent. (See Luke 15:10)
Luke 24:47: and repentance FOR the forgiveness of sins will be preached in his name to all nations, beginning at Jerusalem.
Acts 2:38: Peter replied, “Repent and be baptized, every one of you, in the name of Jesus Christ FOR the forgiveness of your sins. And you will receive the gift of the Holy Spirit.
Acts 8:22: Repent of this wickedness and pray to the Lord in the hope that he may forgive you for having such a thought in your heart.
2 Cor. 12:21: I am afraid that when I come again my God will humble me before you, and I will be grieved over many who have sinned earlier and have not repented of the impurity, sexual sin and debauchery in which they have indulged.
Heb. 6:1: Therefore let us move beyond the elementary teachings about Christ and be taken forward to maturity, not laying again the foundation of repentance from acts that lead to death, and of faith in God,
To conclude, not every verse has “repent from sins” (and so on), but most have just about the exact wording. We need biblical teaching about repentance to repent around the church nowadays.
At that link, look for Mounce.