Repentance simply means a change of mind.
Repentance - G3341 - metanoia
From G3340; (subjectively) compunction (for guilt, including reformation); by implication reversal (of [another's] decision): - repentance.
Godly sorrow works a genuine repentance unto salvation. The godly sorrow produces a fear of God, vehement desires, indignation over sin, a zeal for righteousness, and a clearing (purity in the greek).
For godly sorrow worketh repentance to salvation not to be repented of: but the sorrow of the world worketh death. For behold this selfsame thing, that ye sorrowed after a godly sort, what carefulness it wrought in you, yea, what clearing of yourselves, yea, what indignation, yea, what fear, yea, what vehement desire, yea, what zeal, yea, what revenge! In all things ye have approved yourselves to be clear in this matter. (2Cor 7:10-11)
Jesus said that we must repent or perish.
I tell you, Nay: but, except ye repent, ye shall all likewise perish. (Luk 13:3)
Jesus used Nineveh as an example of repentance.
The men of Nineveh shall rise in judgment with this generation, and shall condemn it: because they repented at the preaching of Jonas; and, behold, a greater than Jonas is here. (Mat 12:41)
So the people of Nineveh believed God, and proclaimed a fast, and put on sackcloth, from the greatest of them even to the least of them. For word came unto the king of Nineveh, and he arose from his throne, and he laid his robe from him, and covered him with sackcloth, and sat in ashes. And he caused it to be proclaimed and published through Nineveh by the decree of the king and his nobles, saying, Let neither man nor beast, herd nor flock, taste any thing: let them not feed, nor drink water: But let man and beast be covered with sackcloth, and cry mightily unto God: yea, let them turn every one from his evil way, and from the violence that is in their hands. Who can tell if God will turn and repent, and turn away from his fierce anger, that we perish not? And God saw their works, that they turned from their evil way; and God repented of the evil, that he had said that he would do unto them; and he did it not. (Jon 3:5-10)
Repentance is the mind change that produces a genuine change of action in a person. A person who repents of their rebellion to God will forsake that rebellion to God.
If they do not forsake that rebellion to God then the repentance was not genuine because they still prefer their rebellion.
Jesus spoke of repentance in Luke 15.
I say unto you, that likewise joy shall be in heaven over one sinner that repenteth, more than over ninety and nine just persons, which need no repentance. (Luke 15:7)
Jesus then went on and gave the parable of the Prodigal Son.
Luke 15
11 And he said, A certain man had two sons:
12 And the younger of them said to his father, Father, give me the portion of goods that falleth to me. And he divided unto them his living.
13 And not many days after the younger son gathered all together, and took his journey into a far country, and there wasted his substance with riotous living.
14 And when he had spent all, there arose a mighty famine in that land; and he began to be in want.
15 And he went and joined himself to a citizen of that country; and he sent him into his fields to feed swine.
16 And he would fain have filled his belly with the husks that the swine did eat: and no man gave unto him.
17 And when he came to himself, he said, How many hired servants of my father's have bread enough and to spare, and I perish with hunger!
18 I will arise and go to my father, and will say unto him, Father, I have sinned against heaven, and before thee,
19 And am no more worthy to be called thy son: make me as one of thy hired servants.
20 And he arose, and came to his father. But when he was yet a great way off, his father saw him, and had compassion, and ran, and fell on his neck, and kissed him.
21 And the son said unto him, Father, I have sinned against heaven, and in thy sight, and am no more worthy to be called thy son.
22 But the father said to his servants, Bring forth the best robe, and put it on him; and put a ring on his hand, and shoes on his feet:
23 And bring hither the fatted calf, and kill it; and let us eat, and be merry:
24 For this my son was dead, and is alive again; he was lost, and is found. And they began to be merry.
The son rebelled against the father and found himself in the pig pen.
He came to himself in the pig pen and changed his mind (Luke 15:17) and decided that he would forsake the pig pen and fall before His father seeking mercy (Luke 15:18).
He then left the pig pen (Luke 15:20) and when he was a long way off his father ran out. The son was broken before him in true sorrow, having forsaken his rebellion, and the father then responded by restoring him (Luke 15:21-24).
This part of this parable is a picture of repentance and is reflected in many other scriptures throughout the Bible.
Acts 3:19
Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord;
Pro 28:13
He that covereth his sins shall not prosper: but whoso confesseth and forsaketh them shall have mercy.
One must repent for sins to be remitted.
Mark 1:4 John did baptize in the wilderness, and preach the baptism of repentance for the remission of sins.
It is through a genuine repentance that one is able to walk in the light because the rebellion has stopped. The blood of Jesus Christ is then applied to the sinner and they are cleansed of all their sin.
But if we walk in the light, as he is in the light, we have fellowship one with another, and the blood of Jesus Christ his Son cleanseth us from all sin. (1 Joh 1:7)
Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; (Rom 3:25)
How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God? (Heb 9:14)