What is the significance of these ten days? 

It is during these days, and only these days, that we implement the final pillar of Islam, Hajj (the pilgrimage to Mecca). By the grace of Allah , my parents performed Hajj last year, and I remember them telling me about the emotional and powerful scene of seeing close to three million people gather from every nook and cranny of the world. People with different skin colors, speaking languages many of us haven't even heard of. Yet one thing united them all, and one thing alone: laa ilaaha ill Allah Muhammad ar-rasool allah (there is no deity but God, and Muhammad is the messenger of God).

When Allah created Adam `alayhi assalaam (peace be upon him), the angels were baffled. They asked Allah , "Are You going to create a creation that will spread corruption and spill blood!? While here we are, worshipping you and praising you" (2:30). They could not understand the wisdom behind a creation that is inclined toward sin, while they are worshipping and praising Allah all the time. The angels do not eat, they do not sleep, and they do not sin. Allah replied, "I know what you do not know" (2:30). During Hajj, on the day of Arafah, Allah calls out to the angels, and points to the pilgrims saying, "(Look,) my slaves have come to Me, looking rough, from every deep valley hoping for My mercy, so if their sins are equivalent to the amount of sand or the drops of rain or like the foam on the sea I will forgive them. So go forth My slaves! Having forgiveness and for what or who you have interceded for" (At-Tabarani). Subhan Allah (glory be to God). Allah is self-subsisting and needs not the worship, nor the praise, nor the sacrifice of us measly human beings, yet one cannot help but feel overwhelmed by emotion at the fact that our Lord is so proud that the pilgrims left their homes, their belongings, their families, and everything for His sake; so proud that He even boasts and shows them off to the angels. Allah knew what they did not know.

The Prophet said, "Whoever does Hajj for the sake of Allah and does not have sexual relations (with his wife), commit sin, or dispute unjustly during the Hajj, will come back like the day his mother gave birth to him" (Bukhari, Muslim).

By receiving the rewards of Hajj & Umrah, without actually performing them:

The Prophet said, "Whoever prays Fajr in congregation then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of Hajj and 'Umrah." He said, "In full, in full, in full." (Tirmidhi)

By constantly making dhikr (remembrance) of Allah and reciting Qur'an:

The Prophet said, "There are not any days that are holier than these ten days of Dhu'l-Hijjah. Therefore increase in doing good deeds to Allah, and increase in saying Allahu Akbar [God is Greater], and saying alhamdulillah [all praise be to God], and saying laa ilaha illAllah [there is no deity but God]" (Ahmed). After hearing this, the sahabah (companions of the Prophet ) became so eager to say these things, that "they would shout them aloud, even in the market places" (Al-Bukhari), after the prayers, and especially after the prayers until three days after Eid (Eid al-Adha, the holiday that falls on the last day of the ten days of Dhu'l-Hijjah). When we truly love something and feel its need, we can't stop thinking about it and craving it. When we haven't eaten for a while, and are hungry, all we think about is food. Similarly, a true mu'min (believer) recognizes how much he needs Allah and just how parched his soul is, and realizes that his soul can only be quenched with the remembrance of Allah . For, "Unquestionably, by the remembrance of Allah hearts are assured," (Qur'an, 13:28).

By increasing our fard (obligatory) & nafl (voluntary) prayers and deeds:

The Prophet said, "Allah said, 'Whoever takes a wali [loyal trustee] of Mine as an enemy, I will wage war on him; and My slave will not perform any act with which he draws closer to Me, more beloved to Me then when he fulfills what I have ordained on him; and My slave will keep drawing closer to Me by performing the nawafil [voluntary acts of worship] until I love him, and when I love him, I will be his hearing with which he hears, his sight with which he sees, his hands with which he strikes, and his legs with which he walks; and if he asks Me, I will give him; and if he seeks refuge with Me, I will grant him refuge.'" (Bukhari)

By giving charity:

The Prophet said, "Verily charity removes the wrath of Allah and eases the sufferings of death." (Tirmidhi) But what about those amongst us who don't have the means to give? The Prophet said, "The reward of giving one dirham has become has surpassed that of giving a hundred thousand dirhams." The companions asked, "How can that be O Messenger of Allah?" He replied, "A rich man takes a hundred thousand dirhams from his wealth and gives it away as Sadaqah. Another man has nothing except two dirhams, and so he takes one dirham and gives it away in charity." (Nasa'i and ibn Hibban who graded it sahh) Subhan Allah, our rabb (Lord) is so merciful and loving that even if you only have a dollar to give, He will accept it, multiply it, and love you for it. And if you can't even give one dollar, then remember that Rasool Allah said, "Your smiling in the face of your brother is charity, your removing of stones, thorns, and bones from people's paths is charity, and your guiding a man gone astray in the world is charity for you." (Tirmidhi)

By spending our nights in tahajjud (late night prayers):

In one of my favorite and most powerful verses in the Qur'an, Allah says, "O My servants who have transgressed against themselves [by sinning], do not despair of the Mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful," (Qur'an 39:53). Allah is giving us hope and reassuring us that no matter how far we've have gone off the path, it's never too late to come back. Who are we, pathetic, measly, human beings to think that our sins are too great in quality and quantity for Allah to forgive!? Who are we to think that our sins are too substantial for the Creator of the heavens and the earth to forgive!? Allah can, and Allah will forgive all of our sins. We are not perfect, we are not angels, and we can never stop sinning. In fact, the Prophet said, "I swear by Him in whose hand is my soul, if you were a people who did not commit sin, Allah would take you away and replace you with a people who would sin and then seek Allah's forgiveness so He could forgive them." (Muslim) Allah loves to forgive us, so we try our best, and when we do slip, we feel guilty, we repent, and promise to never go back to the sin. "O son of Adam! As long as you invoke Me and plead to Me, I will forgive you whatever you have committed, and I will not make much of it. O son of Adam! If your evil deeds reach the borders of the sky, and then you ask Me for forgiveness, I will forgive you. O son of Adam! If you bring forth the earth full of errors, then you meet Me while you do not associate anything (or anyone) with Me, I will bring forth for you its full of forgiveness." (Tirmidhi)

The 10th day of Dhu'l-Hijjah is the greatest day of the year. It is called Yom un-Nahr, the day of the sacrifice, the day of Eid-ul-Adha. The Prophet said: "The best day with Allah is Yawm an-Nahr, the day of sacrifice and then Yawm al-Qarr (the day that follows, when the pilgrims reside in the camps of Mina for worship)." (Ahmad, Abu Dawood)

Also, it is important to remember that not only is it sunnah (Prophetic tradition) to attend Eid salah (prayer), it is obligatory.

Let us put our hand out to Allah , for He will put His arm out to us. Let us go walking towards Allah , for He will come to us running.

Oh Allah! Send Your blessings upon Muhammad and upon the family of Muhammad, as You have sent Your blessings upon Ibrahim and upon the family of Ibrahim . Indeed, You are praiseworthy and glorious.