Forgiveness means to "let someone off the hook". It means to pardon someone for their mistakes towards you.
Forgiveness is a very good deed and shows that you have a big heart.
Allah tells us in Qur'an, Surah 64 ayat 14:
This shows us that if you are kind enough to forgive someone who has done something against you, Allah, who is The Kindest of all will also forgive you.
Prophet Isa (A) said that you should do to others what you would like them to do to you. This means that if you want someone to forgive your sins towards them, you should first forgive their sins towards you.
Imam Ali (A) has said:
"A person who has asked forgiveness from Allah, that person is like a new born baby."
When a baby is born, it is pure and sinless. Everyone loves it because it has done no wrong, and it has harmed no one.
When Allah forgives a person, then that person becomes pure as well. Allah makes his heart and soul as clean as that of a new born baby.
Imam Musa al-Kadhim (A) has said that:
"Your activities are like business with God. If one begs forgiveness for a deed before 7 hours have passed, it is better for you."
If you treat your actions as if Allah counts them, then the minute you make a sin, you should seek forgiveness for it before you finish the day.
That way, when you go to sleep, at least you know that you have settled all accounts with Allah, and so you can sleep happy and at peace with yourself.
(FURTHER REFERENCE; ANECDOTES OF REFLECTION PART 3 CHAPTER 59)