The believers are but brothers, so make peace with your brothers. And fear Allah so that you may receive Mercy.
The Quran 49:10 (Surah al-Hujurat)
Racism is an old issue, yet it is still prevalent in our society today. With the exception of Islam, no other social or religious ideology can cure racism. However, before going any further, it is important for us to understand what racism is.
Simply put, racism is the belief that one race is superior to the other, and the practice of treating a person or group of people differently on the basis of their race. The entire act of racism stems from pride and leads to a superiority complex, based on a person’s color or status.
Of course, pride in itself is against the teachings of Islam, so that rules out any scope for racism in Islam. As reported by Hz Abdullah ibn Masud (RA): 
The Prophet (PBUH) said, “He who has in his heart the weight of a mustard seed of pride shall not enter Paradise.”
It was pride that got Iblis kicked out of paradise. When Allah told him and the angels to bow to Prophet Adam (AS), all the angels agreed, except Iblis. Why? Because he was proud enough to assume that he was better than Adam, for he was created from fire, whereas Hz Adam (AS) was created from clay.
Obviously, just as Iblis could not claim superiority on the basis of fire vs clay, how can we claim that one person is better than the other simply due to the color of one’s skin? At the end of the day, we are judged not by the color of our skin, but by the nature of our actions and deeds. Unfortunately, many Muslims too are nowadays forgetting the teaching of their own religion, and behaving as racists.
There are numerous verses in the Quran that speak about brotherhood or sisterhood, and renounce racism, such as: 
Verily, this nation of yours is one nation and I am your Lord, so worship Me.
Hold firmly to the Rope of Allah all together and do not become divided. Remember the Favor of Allah upon you, when you were enemies and He brought your hearts together and you became brothers by His Favor. You were on the edge of a pit of the Fire and He saved you from it. Thus does Allah make clear to you His verses that you may be guided.
Racism existed in human society since times immemorial. Allah sent us His Message in the Quran, and abolished ignorant and evil practices such as racism. Thus, racism is not permissible in Islam, and is against the very nature of Islamic teachings.
In fact, in the last sermon of Prophet Muhammad (PBUH), he reminded us: 
O People, all mankind is from Adam and Eve; an Arab has no superiority over a non-Arab nor a non-Arab has any superiority over an Arab; also a white has no superiority over a black, nor a black has any superiority over a white, except by piety and good action.
The above words of our Beloved Prophet (PBUH), in fact, highlight the very essence of Islam. We are one community, and it is obligatory for us to stand together, and treat everyone with love and respect. Artificial factors such as color, caste, wealth, beauty, region, etc. should not become a barrier between us. It doesn’t matter what part of the world you live in — Arabia, Somalia, India & Pakistan, China, Japan, Nigeria, Bosnia, Turkey, Brazil or Venezuela. Nor does it matter what the color of your skin is — white, brown, yellow or black. We have all been created by Allah, and as Muslims, we should always be aware of the fact that there is no scope for racism in Islam.
Lastly, Allah has created us in unique ways for us to understand that regardless of where we are from, or what color we are, or which language we speak, we all are still brothers and sisters in this faith. I will now end my article with this beautiful verse from the Quran: 
O mankind, indeed we have created you from male and female, and made you peoples and tribes so that you may know one another. Indeed, the most noble of you in the Sight of Allah is the most righteous of you. Indeed, Allah is Knowing and Acquainted.
- Sahih Muslim Book 01 Hadith 166
- The Quran 21:92 (Surah al-Anbiya)
- The Quran 03:103 (Surah al-Imran)
- The Final Sermon of Prophet Muhammad (PBUH)
- The Quran 49:13 (Surah al-Hujurat)