Assalam Alaikum,
As Muslims, (we should all be of the understanding) are taught that if Allah can forgive our sins even if they are sooooooo much and there is a hadith that says:
"Whoever says, 'Subhan Allah 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," Volume 8, Book 75, Number 414, Narrated Abu Huraira,
so how then can any Muslim say that they have the right not to forgive anyone when Allah is willing to forgive so much? While we cannot come even close to God's mercy, we need to try to be good Muslims as if this is a quality of Allah then surely it is one that Allah/ would like to see in us. As Muslims, we are going to make mistakes, just becuase we have a perfect religion doesn't mean that all Muslims will be perfect, and so for some people it is hard to as is often said "forgive and forget", but Allah knows that for us to forgive others is the best thing, and we are often taught in the Quran and Sunnah that we should have patience (sabr), and forgive others and indure difficulties.
Inshallah, here are some material that will help you in understanding true Islamic principles of forgiveness:
this is a good summary article:
http://www.geocities.com/mutmainaa/tafakkur/forgive_hate.html
and please also read these items below:
Better and more rewarding is God's reward to those who believe and put their trust in Him: who avoid gross sins and indecencies and, when angered, are willing to forgive... Let evil be rewarded by like evil, but he who forgives and seeks reconciliation shall be rewarded by God. He does not love the wrongdoers.... True constancy lies in forgiveness and patient forbearance.
Qur'an 42.36-43
'Abdullah b. Umar said that Rasulullah saw said : "The muslim is he from whose tongue and hand a Muslim is safe, and the muhajir he who gives up what Allah has prohibited for him ". ( Bukhari, Muslim )
Jabir b. Abdullah said that Rasulullah saw said : "Allah is not merciful to whim who is not merciful to people ". ( Bukhari, Muslim )
Abdullah b. Umar said that Rasulullah saw said :"The Merciful One shows mercy to those who are themselves merciful (to others). So show mercy to whatever is on earth, then He who is in heaven will show mercy to you " ( Abu Dawud, Tirmidhi )
From Abu Hurairah : Rasulullah saw said : "He who does not thank people, does not thank Allah". ( Ahmad, Tirmidhi
From Anas : Rasulullah saw said : "By Him in whose hand is my soul, a servant (of Allah) does not believe (truly) until he likes for his brother what he likes for himself ". ( Bukhari, Muslim )
From Abu Ayyub al-Anshari : Rasulullah saw said : " It is not right for a man to abandon his brother for more than three days". ( Bukhari, Muslim )
Jabir said that Rasulullah saw said : " If one makes excuses to his brother, but he does not excuse him, or accept his apology, he is as sinful as one who takes an unjust tax " ( Baihaqi )
From Abu Hurairah : Rasulullah saw said : " The strong man is not the one who is strong in wrestling, but the one who controls himself in anger " ( Bukhari, Muslim )
From Ibn Umar ra : Rasulullah saw said : "The muslim who meets with people and endures any harm they may do is better than he who does not mix with them and does not endure any harm they may do". ( Tirmidhi )
Ibn Umar ra narrated Rasulullah saw said : "O you who declare Islam with your tongues but whose hearts have not been reached by faith, do not annoy the Muslims nor seek out their faults, for he who seeks out the faults of his brother Muslim will have his faults sought out by Allah, and when Allah seeks out someone's faults, He exposes them, eventhough he should be in the interior of his house." ( Bukhari, Muslim )
Abu Dharr ra said that Rasulullah s.a.w said to him : "Fear God wherever you are; let an evil deed (be) followed by a good deed so that you blot it out; and be well-behaved towards people". ( Ahmad, Tirmidhi )
Abu Hurairah ra said that Rasulullah s.a.w said : "He who truly believes in Allah and the last Day, should speak good or keep silent". ( Bukhari, Muslim )
Al-Nour (The Light)
24:22 Hence, [even if they have been wronged by slander,] let not those of you who have been graced with God’s favour and ease of life ever become remiss in helping [25] [the erring ones among] their near of kin, and the needy, and those who have forsaken the domain of evil for the sake of God, [26] but let them pardon and forbear. [For,] do you not desire that God should forgive you your sins, seeing that God is much-forgiving, a dispenser of grace? [27]
I posted more hadith than Quran, because to really look at forgiveness in the Quran, I would have to post the whole thing! Islam strongly emphasizes this, so remember always, that Islam is a complete and full religion that has been perfected and from which nothing is lacking. Alhumdulillah!