The Quran exhorted Muhammad and his followers to forgive those who did not believe in the days of Allah:
Say (O Muhammad SAW) to the believers to forgive those who (harm them and) hope not for the Days of Allah (i.e. His Recompense), that He may recompense people according to what they have earned (i.e. to punish these disbelievers, who harm the believers).
Whosoever does a good deed, it is for his ownself, and whosoever does evil, it is against (his ownself). Then to your Lord you will be made to return. S. 45:14-15 Hilali-Khan
In the immediate context it seems to be certain that by "the Days of Allah" the author meant the time when Allah will bring all mankind into judgment in order to repay them for what they have said and done. At the very least the last day is naturally included within this phrase since this happens to be one of Allah’s days.
Hence, the meaning of the text is quite straightforward: Muslims are to forgive and not harass anyone who denies that there is a last day in which Allah will judge humanity.
But this is expressly contradicted by the following command to Muslims to fight any and every one who does not believe in this last day!
Fight those who believe not in God nor the Last Day, nor hold that forbidden which hath been forbidden by God and His Apostle, nor acknowledge the religion of Truth, (even if they are) of the People of the Book, until they pay the Jizya with willing submission, and feel themselves subdued. S. 9:29 Y. Ali
Note the glaring contradiction:
Not surprisingly, Muslim expositors had to resort to the doctrine of abrogation to explain away this obvious discrepancy:
(Tell) O Muhammad (those who believe) 'Umar and his fellow believers (to forgive those who hope not for) those who fear not (the days of Allah) the torment of Allah, the reference here is to the people of Mecca; (in order that He may requite folk) i.e. 'Umar and his fellow believers (what they used to earn) what they used to do of good deeds. This forgiveness relates to the period before the migration to Medina. They were later commanded to fight the disbelievers. (Tanwîr al-Miqbâs min Tafsîr Ibn ‘Abbâs; source; bold and underline emphasis ours)
Tell those who believe to forgive those who do not anticipate, [those who do not] fear, the days of God, the events which He precipitates, that is to say, forgive the disbelievers the harm which they have caused you - this was before the command to struggle against them [had been revealed] - that He, God, may requite (a variant reading [for li-yajziya] has li-najziya, 'that We may requite') a people for what they used to earn, when they used to forgive disbelievers the harm which they did. (Tafsir al-Jalalayn; source; bold and underline emphasis ours)
Allah's saying;
<Say to the believers to forgive those who hope not for the Days of Allah,> means, let the believers forgive the disbelievers and endure the harm that they direct against them. In the beginning of Islam, Muslims were ordered to observe patience in the face of the oppression of the idolators and the People of the Scriptures so that their hearts may incline towards Islam. However, when the disbelievers persisted in stubbornness, Allah legislated for the believers to fight in Jihad. Statements of this meaning were collected from `Abdullah bin `Abbas and Qatadah. Mujahid said about Allah's statement,
<those who hope not for the Days of Allah,> "They do not appreciate Allah's favors." Allah said,
<that He may recompense people, according to what they have earned.> meaning, if the believers forgive the disbelievers in this life, Allah will still punish the disbelievers for their evil in the Hereafter. Allah's statement next,
<Whosoever does a good deed, it is for himself, and whosoever does evil, it is against (himself). Then to your Lord you will be made to return.> meaning, you will all return to Allah on the Day of Resurrection, when you and your actions will be displayed before Him. Then, He will recompense you for your deeds, good for good and evil for evil. (Tafsir Ibn Kathir, English version, abridged by a group of scholars under the supervision of Shaykh Safiur-Rahman Al-Mubarakpuri: bold and underline emphasis ours)
The above explanations are simply another way of saying that Muhammad’s circumstances determined and dictated what Allah would send down, to the extent that Allah even changed his mind and gave commands that contradicted one another! After all, when Muhammad was in Mecca and greatly outnumbered by the unbelievers Allah told Muslims to be tolerant and forgiving. But when Muhammad migrated to Medina and amassed enough power Allah changed his mind and decided to command Muslims to fight those who disbelieved in Islam and the last day!
To put it simply, abrogation is nothing more than the Islamic way of saying that the Quran contains egregious mistakes and blatant contradictions.
Further Reading
Should Muslims Accept Peace or Not? — Islam and Warfare
Contradictions in the Qur'an
Answering Islam Home Page