The Lexicographic Interpretation of Sin and Forgiveness:

Sin can be defined in Islam as follows:
- Breaking of the law of God on purpose or the state or condition resulting from this act.
- Wrong doing of any kind; immoral act.
- A violation of any Islamic rule or standard, as
of taste or propriety.

Forgiveness can be defined in Islam as follows:
- To give up the wish to punish or get even with an enemy or offender.
- Not have hard feelings at or towards others.
- To give up all claims for damage inflicted on
you by others.
According to Islam, to err or commit a sin is human, to forgive is divine.

The Origin of Sins in Islam:
Islamic law urged man to be on his guard against sinning, and it sets forth a number of restraints, indications, precautions, and impediment that would prevent people from falling into those sins threatening the safety of the whole society.
Sins are numerous and manifold. The most dangerous type of which comprises those grave or deadly sins which are not revealed (hidden in the heart) such as idolatry, atheism, heresy, hypocrisy, envy, malice, vanity, arrogance, conceit, and cheating. Sins can be related to negligence of one's obligations towards Allah. And since such sins are between man and Allah, they are more hopefully expected to be forgiven than others. Sins can also be related to one's transgressing on other people's rights. Here, punishment or absolutions are induced during a sinful person's life.

Sins are divided into grave sins and venial sins. Grave sins are concerned, with transgresses on the rights of Allah, individuals and society.
The number of grave sins is indefinite; however, it can be confined to those sins whose perpetrators are promised Hell.
Allah says in the Holy Qur'an, "If anyone contends with the Messenger even after guidance has been plainly conveyed to him, or follows a path other than that becoming to men to faith, We shall leave him in the path he has chosen,
and land him in Hell, - What an evil refuge!"
If a believer avoids these deadly sins, he would be forgiven the venial ones. This is emphasized in the Holy Qur'an where
Allah says "If ye (but) eschew the most heinous of thing which ye are forbidden to do, we shall remit your
evil deeds and admit you to a Gate of great honor."
However, venial sins can turn into grave ones if they are scornfully underestimated and constantly committed.
Abdullah bin Mas'ud related to us one narration saying, "A believer sees his sins as if he were sitting under a mountain which, he is afraid, may fall on him; whereas a wicked evil-doer considers his sins as flies passing over his nose and he just drives them away like this." (Abu Shihab, the sub-narrator, moved his hand over his nose in illustration).

Forgiving Sins through Observing Religious Duties, Following the Sunna of the Prophet, and Performing Desirable Acts (those acts whose neglect is not punished by Allah, but whose performance is rewarded):

Allah has decreed some religious duties which are classified into articles of faith, observances and mutual relations. Performing these duties is greatly rewarded especially such observances as prayer, fasting, alms-giving and pilgrimage which are regarded as atonement for a Muslim's sins. Abu Huraira narrated that the Prophet (peace be upon him) said, "The prayer offered in congregation is seventy-five times more superior (in reward) to the prayer offered alone in one's house or in a business center, because if one performs ablution and does it perfectly, and then proceeds to the mosque with the sole intention of offering prayer, then for each step which he takes towards the mosque, Allah upgrades him a degree in reward and (forgives) crosses out one of his sins (at each step) till he enters the mosque. He is considered in (prayer) as long as he is waiting for the prayer, and the angels keep on asking for Allah's forgiveness for him, and they keep on saying: "O Allah! Forgive him, as long as he keeps on sitting at his praying place and does not pass wind."
Also Abu Huraira narrated that the Prophet (peace be upon him) said, "Whoever fasts during the month of Ramadan out of sincere faith, and hoping to attain Allah's Rewards, then all his past sins will be forgiven." Abu Huraira narrated that the Prophet (peace be upon him) said: "(The performance of) Umra is an expiation for the sins committed between it and the previous one. And the reward of Hajj (the one accepted by Allah) is nothing but Paradise."

Following the Sunna of the Prophet (peace be upon him) and performing desirable acts are one way of atoning for sins.
Allah says in the Holy Qur'an, "... for those things that are good remove those that are evil ..."
Having patience (forbearing) with afflictions and misfortunes, such as illness, poverty, etc., is also rewarded.
Abu Said Al-Khudri and Abu Huraira narrated that the Prophet (peace be upon him) said, "No fatigue, no disease, nor sorrow, nor sadness, nor hurt, nor distress befalls a Muslim, even if it were the prick he receives from a thorn, but Allah expiates some of his sins for that." And Abdullah narrated that I visited the Prophet (peace be upon him) during his ailments and he was suffering from high fever. I said, "You have a high fever. Is it because you will have a double reward for it?" He said, "Yes, for no Muslim is afflicted with any harm but for that Allah will remove his sins as the leaves of a tree fall down." Abu Huraira also narrated: The Prophet (peace be upon him) said, "If Allah wants to do good to somebody, He afflicts him with trials."

Forgiving Sins through Turning to Allah in Repentance and Seeking His Pardon:

Allah is Much-Forgiving; He is the only One entitled to accepting repentance and to forgiving sins. He says in the Holy Qur'an,"O ye who believe! Turn to Allah with sincere repentance: in the hope that your Lord will remove from you your evil deeds, and admit you to Gardens beneath which Rivers flow."
And He says, "Know they not that Allah doeth accept repentance from His votaries?"
Abu Huraira narrated that I heard Allah's Messenger (peace be upon him) saying, "By Allah! I seek Allah's Forgiveness and turn to Him in repentance for more than seventy times a day."
The heart of man, as created by Allah, is pure and unsullied. Every time that a man does an ill deed, it marks a stain or rust on his heart. But on repentance and forgiveness, such stain is washed off. If there is no repentance and forgiveness, the stains deepen and spread more and more, until the heart is sealed, and eventually the man dies a spiritual death.
Due to human weaknesses, Allah urges all Muslims to seek His forgiveness and to turn to Him in repentance. He says in the Holy Qur'an, "And seek ye the Grace of Allah: for Allah is Oft-Forgiving, Most Merciful." And He says, "If anyone does evil, or wrongs his own soul but afterwards seeks Allah's forgiveness, he will find Allah Oft-Forgiving, Most Merciful."

When sins concern one's obligations and duties towards Allah are committed, repentance can only be accepted of those who meet the following provisions: feeling regretful, refraining from committing more sins, and having firm resolution not to commit such sins again. And when sins are related to one's transgression upon somebody else's right, the transgressor will have to repair what he has imparted, and compensate affected person for the damage done until, by fair means, both sides reach a mutual consent.

Forgiving Sins through Supplication and Invocation of Allah:

As this life is not the end of all things, and we are to have its fulfillment in the Hereafter, we have only to pray to the Lord of the Present and the Hereafter, and He will hear us, forgive us, guide us, and make our path smooth. But pride will have its fall, and its humiliating Punishment. Allah says in the Holy Qur'an, "And your Lord says: Call on Me; I will answer your (prayer): but those who are too arrogant to serve Me will surely enter Hell Abased." [color=#000000]He also says, "When My servants ask thee concerning Me, I am indeed close (to them): I respond to the prayer of every suppliant when he calleth on Me."[/color]
The Prophet (peace be upon him) used to urge the Muslims to call on Allah and pray to Him. A great many Hadiths [Prophetic traditions or narratives relating deeds and utterances of the Prophet (peace be upon him)] support this idea Supplication cannot be sincere unless Allah is always remembered. Allah says in the Holy Qur'an, "Then do ye remember Me; I will remember you. ", and "For men and women who engage much in Allah's remembrance, Allah has prepared forgiveness and great reward."
Invocation of Allah purifies the heart and protects it from worldly temptations which can drive a seduced person into perilous situations. Invocation of Allah crosses out sins and upgrades a suppliant's rank in the Hereafter.
Abu Huraira narrated that the Prophet (peace be upon him) said, "Whoever says, 'Subhan Allahi Wa bihamdihi' one hundred times a day, will be forgiven all his sins even if they were as much as the foam of the sea." 'Subhan Allahi wa bihamdihi' means 'I deem Allah free of any resemblance to anything whatever in any respect, and I glorify His praises.'
Since one's invocation must be accepted by Allah, a Muslim has to show patience, has to appeal to Allah with determination, and he should not lose hope if the response to his invocation is delayed. Abu Huraira narrated that the Prophet (peace be upon him) said, "None of you should say: 'O Allah! Forgive me if You wish; O Allah, be Merciful to me if You wish, but he should always appeal to Allah with determination, for nobody can force Allah to do something against His Will"; and he (peace be upon him) said, "The invocation of anyone of you is granted (by Allah) if he does not show impatience (by saying, I invoked Allah but my request has not been granted')". A Muslim will have to refrain from earning his living dishonestly, for Allah does not accept the invocation of those who encroach upon others' rights.
Abu Huraira narrated that the Prophet (peace be upon him) said, "If one gives in charity what equals one date-fruit from the honestly earned money-and Allah accepts only the honestly earned money- Allah takes it in His Right (hand) and then enlarges its reward for that person (who has given it), as anyone of you brings up his baby horse, so much so that it becomes as big as a mountain."
In conclusion, we can say that the Islamic concept of sin and its means of forgiveness are very simple. It is quite easy to comprehend and practice. Adam and his wife Eve had committed a sin by eating from the forbidden tree. But they have repented and asked forgiveness from Allah who accepted their repentance. Hence, there was no need to assume that the original sin committed by them was inherited by their innocent children.
According to Islam committing a sin or going stray is an individual choice. Seeking forgiveness is an individual responsibility. There are so many ways, a Muslim can attain forgiveness from Allah for his sins. These include abstaining from forbidden things especially grave sins, observing religious duties, following the sunna of the Prophet (peace be upon him), performing desirable acts, turning to Allah in repentance and seeking his pardon, reciting the Holy Qur'an, supplication and invocation of Allah at all times......


Summarised from https://www.dropbox.com/s/7iuzfjpybgzw0d8/english_Sin_and_Forgiveness_in_Christianity_and_Islam.pdf?dl=0 By Dr. Ezzia Ali Taha

Will be adding the quotations references later inshaAllah

May Allah increase us in humility and righteousness, and forgive our sins.

Good one... Jazak Allah Khairan!

9jaforlife:
May Allah increase us in humility and righteousness, and forgive our sins.

Good one... Jazak Allah Khairan!

Ameeen. Wa iyyak