What is Zulhijjah?
Zulhijjah (or Dhulhijjah) is the last month of the Islamic calendar. Literally, it means "hajj." Obviously, this name of the month indicates that the great annual worship of "hajj" is performed in this month, which gives it special significance.
Some specific merits and rules relevant to this month:
1. Allah has taken oath of ten nights in Surah Al-Fajr. According to the majority of the commentators of Qur’ãn, the nights are those of the (first) ten days of Zul Hijjah.
2. Abdullah Ibne ‘Abbãs radiyallahu anhu related that the Prophet sallalahu alayhi wasallam said, “Good deeds performed on other days are not superior to those performed on these (first ten days of Zul Hijjah).” The Companions radiyallahu anhum inquired, “Not even Jihãd?” He replied, “Not even jihãd, except for that person who goes out putting himself and his wealth in danger and does not return with anything.” (Bukhãri)
3. Abdullah Ibne ‘Abbãs radiyallahu anhu narrated that Rasulullah sallalahu alayhi wasallam said, “On no other days are good deeds more liked by Allah than on these ten days (i.e. The first ten days of Zul Hijjah).” The Sahãbah radiyallahu anhum asked, ” O Rasulullah ! Not even Jihãd in the Way of Allah?” Rasulullah sallalahu alayhi wasallam replied, “Not even Jihãd in the Way of Allah, except for that person who goes out with his life and wealth and does not return with anything.” (Abu Dãwud, Tirmidhi, Ibne Mãjah)
4. Abu Hurayrah radiyallahu anhu related that Rasulullah sallalahu alayhi wasallam said, “On no days is the worship of Allah desired more than in the (first) ten days of Zul Hijjah. The fast of each of these days is equal to the fast of a whole year, and the worship of each of these nights is equal to the worship of Laylatul Qadr.” (Tirmidhi, Ibne Mãjah)
5. The mother of the believers, Hafsah radiyallahu anhaa reports that Rasulullah sallalahu alayhi wasallam used to fast the (first) nine days of Zul Hijjah. (Nasã’i, Ahmad, Abu Dãwud)
6. Abdullah Ibne ‘Umar radiyallahu anhu related that Rasulullah sallalahu alayhi wasallam said, “No days are as weighty with Allah and so liked by Him for good deeds than the first ten days of Zul Hijjah. So on these days increasingly read Subhãnallah, Lã ilãha illallãh, Alhamdulillah and Allahu Akbar.” (Musnad Ahmad)
The 9th day of Zulhijjah is called 'Youmul - "Arafah' (The Day of 'Arafah). This is the date when the Hujjaj (Haji pilgrims, plural of Haajj) assemble on the plain of 'Arafat, six miles away from Makkah al-Mukarramah, where they perform the most essential part of the prescribed duties of hajj, namely, the 'Wuqoof of'Arafat (the stay in 'Arafat).
For those not performing hajj, it is mustahabb (desirable) to fast on this day according to their own calendar. It sometimes occurs that 9th Zuihijjah falls on different days in different countries according to the sighting of the moon. In such cases, Muslims of each country should observe 'Youmul 'Arafah according to the lunar dates of their own country.
For example, if 'Youmul 'Arafah' is being observed in Saudi Arabia on Friday, and in Pakistan on Saturday, Pakistani Muslims should treat Saturday as 'Youmul 'Arafah' and should fast on that day if they desire to benefit from the fast of 'Youmul'Arafah'.
The fast of 'Youmul 'Arafah' has been emphasized by the Holy Prophet, Sall-Allahu alayhi wa sallam, as a mustahabb (desirable) act.
Abu Qatada said "The Messenger of Allah, may Allah bless him and grant him peace, was asked about fasting the Day of 'Arafah. He said, 'It expiates the (sins of) past year and coming year." [Muslim]
Beginning from the Fajr of the 9th Zulhijjah up to the 'Asr prayer of the 13th, it is obligatory on each Muslim to recite the Takbir of Tashriq after every fardh Salaah in the following words:
Allahu Akbar, Allahu Akbar,
La Ilaha Illallahu, Wallahu Akbar,
Allahu Akbar wa lillahilhamd.
(There is no god but Allah and Allah is the greatest, Allah is the greatest and to Allah belongs all praise.)
According to authentic Islamic sources, it is obligatory on each Muslim male, to recite this Takbir after every fard salah. For women also, it is commendable though not obligatory. Whether you are performing salah with Jama'ah (collectively) or on your own (individually) makes no difference. You must recite the Takbir. However, male Muslims should recite it in a loud voice, while females should recite it in a low voice.
Also see this for detailed info on what to do on these special days (is based on Lataif wa Maarif of Imam Ibn Rajab) http://tazkiyatips.blogspot.com/2012/10/what-to-do-on-blessed-days-of-dhu-l.html