Repentance, in its literal meaning, is to turn back or to direct oneself. In the context of the Quran, repentance means turning towards Allah, and when used in relation to Allah, it means accepting the repentance of the servant. One of Allah's names/attributes is Tawwab, meaning the One who accepts repentance abundantly, and one of the servant's attributes is Ta'ib, meaning the one who repents.
In Raghib al-Isfahani's work al-Mufradat, repentance is mentioned as follows:
Repentance is the most beautiful way to abandon sin. This is the most effective way to apologize. Because there are three forms of apology. The first is for the one apologizing to say, "I didn't do it." The second is to say, "I did it because of this." The third is to say, "I did it, I acted wrongly, but now I have turned back from it/given it away." There is no fourth form. This last one is called repentance. According to Sharia, repentance is to abandon sin because of its wrongdoing, to regret what one has done, to make a firm decision not to return to it, and to correct one's actions/deeds that can be corrected by turning back. When these four conditions come together, the conditions of repentance are fulfilled.
The Quran does not permit anyone, including prophets, to repent on behalf of another. Thus, the Quran does not accept the practice of one person acting as an intermediary to atone for the sins of another, as is the case in religious orders or Christianity. No one can bear the sins of another, nor can they be held responsible for the sins of another.
And no bearer of burdens shall bear another's burden, and if one heavily laden calls another to (bear) his load, nothing of it will be lifted even though he be near of kin. You can warn only those who fear their Lord unseen, and perform As-Salat (Iqamat-as-Salat). And he who purifies himself (from all kinds of sins), then he purifies only for the benefit of his ownself. And to Allah is the (final) Return (of all).
Therefore, no one can repent on behalf of another. Even the Last Prophet was not granted such a right. At the end of the verses concerning the Battle of Uhud, Allah Almighty addresses the Prophet with the following verse, stating that he had no involvement, influence, authority, or responsibility in this matter.
Not for you is the decision; whether He turns in mercy to (pardons) them or punishes them; verily, they are the Zalimun (polytheists, disobedients, and wrong-doers, etc.).
The following verses demonstrate that it is necessary for a person to repent directly and solely to God.
Will they not repent to Allah and ask His Forgiveness? For Allah is Oft-Forgiving, Most Merciful.
O you who believe! Turn to Allah with sincere repentance! It may be that your Lord will remit from you your sins, and admit you into Gardens under which rivers flow (Paradise) the Day that Allah will not disgrace the Prophet and those who believe with him, their Light will run forward before them and with (their Records Books of deeds) in their right hands they will say: "Our Lord! Keep perfect our Light for us and grant us forgiveness. Verily, You are Able to do all things."
"And O my people! Ask forgiveness of your Lord and then repent to Him, He will send you (from the sky) abundant rain, and add strength to your strength, so do not turn away as Mujrimun (criminals, disbelievers in the Oneness of Allah)."
And to Thamud (people, We sent) their brother Salih (Saleh). He said: "O my people! Worship Allah, you have no other Ilah (God) but Him. He brought you forth from the earth and settled you therein, then ask forgiveness of Him and turn to Him in repentance. Certainly, my Lord is Near, Responsive."
And : "Seek the forgiveness of your Lord, and turn to Him in repentance, that He may grant you good enjoyment, for a term appointed, and bestow His abounding Grace to every owner of grace. But if you turn away, then I fear for you the torment of a Great Day (i.e. the Day of Resurrection).
The Forgiver of sin, the Acceptor of repentance, the Severe in punishment, the Bestower (of favours),
La ilaha illa Huwa (none has the right to be worshipped but He), to Him is the final return.
"And ask forgiveness of your Lord and turn unto Him in repentance. Verily, my Lord is Most Merciful, Most Loving."
And He it is Who accepts repentance from His slaves, and forgives sins, and He knows what you do.
But whoever repents after his wrongdoing, and reforms(does righteous good deeds), then Allah shall accept his repentance. Indeed Allah is All-Forgiving, All-Merciful.
The word used in the below verses, translated as "the One who accepts repentance," is Tawwab, which is also one of Allah's attributes. In simpler terms, this shouldn't be interpreted as implying that there are other authorities who accept repentance.
Glorify your Lord with praise and ask Him for forgivenes. He accepts repentance.
Remember, Moses said: "My people, by taking this calf you have done yourselves harm, so now turn to your Creator in repentance, and kill your pride(enfusekum-egos), which is better with your Lord." And (the Lord) softened towards you, for He is all-forgiving and merciful.
Were it not for Allah’s favor upon you, and His mercy, and that Allah is Acceptor of Repentance and Wise.
O you who believe! Avoid much suspicions, indeed some suspicions are sins. And spy not, neither backbite one another. Would one of you like to eat the flesh of his dead brother? You would hate it. And fear Allah. Verily, Allah is the One Who accepts repentance, Most Merciful.
Then Adam received from his Lord [some] words, and He accepted his repentance. Indeed, it is He who is the Accepting of repentance, the Merciful.
Except those who repent and do righteous deeds, and openly declare (the truth which they concealed).
These, I will accept their repentance. And I am the One Who accepts repentance, the Most Merciful.
"Our Lord! And make us submissive unto You and of our offspring a nation submissive unto You, and show us our Manasik (our ways of worship), and accept our repentance. Truly, You are the One Who accepts repentance, the Most Merciful.
Istighfar , a type of prayer, means "seeking forgiveness." It is often confused with repentance (tawbah). Since istighfar is a prayer, believers can pray for each other to receive Allah's forgiveness. Naturally, the Prophet can also pray for forgiveness for other believers.
We sent no Messenger, but to be obeyed by Allah's Leave. If they (hypocrites), when they had been unjust to themselves, had come to you and begged Allah's Forgiveness, and the Messenger had begged forgiveness for them: indeed, they would have found Allah All-Forgiving (One Who accepts repentance), Most Merciful.
So know that La ilaha ill-Allah (none has the right to be worshipped but Allah), and ask forgiveness for your sin, and also for believing men and believing women. And Allah knows well your moving about, and your place of rest.
People can seek forgiveness for others; however, no one can repent for another, because seeking forgiveness is a prayer, while repentance is an act of devotion that the individual must undertake themselves.
And those who came after them say: "Our Lord! Forgive us and our brethren(brothers) who have preceded us in Faith, and put not in our hearts any hatred against those who have believed. Our Lord! You are indeed full of kindness, Most Merciful.
"My Lord! Forgive me, and my parents, and him who enters my home as a believer, and all the believing men and women. And to the Zalimun (polytheists, wrong-doers, and disbelievers, etc.) grant You no increase but destruction!"
"Our Lord! Forgive me and my parents, and (all) the believers on the Day when the reckoning will be established."
Musa (Moses) said: "O my Lord! Forgive me and my brother, and make us enter into Your Mercy, for you are the Most Merciful of those who show mercy."
Allah does not task any soul beyond its capacity. Whatever [good] it earns is to its own benefit, and whatever [evil] it incurs is to its own harm. ‘Our Lord! Take us not to task if we forget or make mistakes! Our Lord! Do not place upon us a burden as You placed on those who were before us! Our Lord! Do not lay upon us what we have no strength to bear! Excuse us, forgive us, and be merciful to us! You are our Master, so help us against the faithless lot!’
The verses in which angels also pray for forgiveness for people are listed below.
Those who bear the Throne, and those who are around it, celebrate the praise of their Lord and have faith in Him, and they plead for forgiveness for the faithful: ‘Our Lord! You embrace all things in mercy and knowledge. So forgive those who repent and follow Your way and save them from the punishment of hell.
The heavens are about to be rent apart from above them, while the angels celebrate the praise of their Lord and plead for forgiveness for those [faithful] who are on the earth. Indeed, Allah is the All-forgiving, the All-merciful!