Ibn 'Abbas says about the verse, "Remember Allah during the well known days," that it refers to the first ten days of Dhul-Hijjah. [This is related by al-Bukhari]
Allah swears an oath by them, and swearing an oath by something is indicative of its importance and great benefit. Allah says (interpretation of the meaning):
"By the dawn; by the 10 nights" - [Noble Quran 89:1-2]
Ibn Abbas, Ibn al-Zubayr, Mujahid and others of the earlier and later generations said that this refers to the first ten days of Dhul-Hijjah. Ibn Kathir said: "This is the correct opinion." [Tafsir Ibn Kathir, 8/413]
Praise be to Allah Who has created Time and has made some times better than others, some months and days and nights better than others, when rewards are multiplied many times, as a mercy towards His slaves. This encourages them to do more righteous deeds and makes them more eager to worship Him, so that the Muslim renews his efforts to gain a greater share of reward, prepare himself for death and supply himself in readiness for the Day of Judgment.
This season of worship brings many benefits, such as the opportunity to correct ones faults and make up for any shortcomings or anything that one might have missed. Every one of these special occasions involves some kind of worship through which the slaves may draw closer to Allah, and some kind of blessing though which Allah bestows His favor and mercy upon whomsoever He will. The happy person is the one who makes the most of these special months, days and hours and draws nearer to his Lord during these times through acts of worship; he will most likely be touched by the blessing of Allah and will feel the joy of knowing that he is safe from the flames of Hell. [Ibn Rajab, al-Lataif, p.8]
Ibn 'Abbas reports that the Messenger of Allah (peace be upon him) said, "No good deeds done on other days are superior to those done on these days [meaning the ten days of Dhul-Hijjah]." The companions asked, "O Messenger of Allah, not even jihad in the way of Allah?" He said, "Not even jihad, except for the man who puts his life and wealth in danger [for Allah's sake] and returns with neither of them." [This is related by the group except Muslim and an-Nasa'i]
Ahmad and at-Tabarani record from Ibn 'Umar that the Messenger of Allah (peace be upon him) said, "There is no day more honorable in Allah's sight and no acts more beloved therein to Allah than those in these ten days. So say tahlil (There is no deity worthy of worship but Allah: La ilaha ill Allah), takbir (Allah is the greatest: Allahu Akbar) and tahmid (All praise is due to Allah: alhumdulillah) a lot [on those days]." [Reported by Ahmad, 7/224; Ahmad Shakir stated it is Sahih]
Abu Hurairah relates that the Messenger of Allah (peace be upon him) said, "There are no days more loved to Allah for you to worship Him therein than the ten days of Dhul Hijjah. Fasting any day during it is equivalent to fasting one year and to offer salatul tahajjud (late-night prayer) during one of its nights is like performing the late night prayer on the night of power. [i.e., Lailatul Qadr]." [This is related by at-Tirmidhi, Ibn Majah, and al-Baihaqi]
Ibn 'Umar narrated that at Mina, the Messenger of Allah (peace be upon him) said, "Do you know what is the day today?" The people replied, "Allah and His Messenger know it better." He said, "It is the forbidden (sacred) day. And do you know what town is this?" They replied, " Allah and His Messenger know it better." He said, "This is the forbidden (sacred) town (Mecca). And do you know which month is this?" The people replied, "Allah and His Apostle know it better." He said, "This is the forbidden (sacred) month." The Messenger added, "No doubt, Allah made your blood, your properties, and your honour sacred to one another like the sanctity of this day of yours in this month of yours in this town of yours."
Narrated Ibn 'Umar: On the Day of Nahr (10th of Dhul-Hijjah), the Messenger (peace be upon him) stood in between the Jamrat during his Hajj which he performed (as in the previous Hadith) and said, "This is the greatest Day (i.e. 10th of Dhul-Hijjah)." The Messenger (peace be upon him) started saying repeatedly, "O Allah! Be Witness (I have conveyed Your Message)." He then bade the people farewell. The people said, "(This is Hajjat-al-Wada)." [Bukhari 2.798]