Ramadan - A Blessed Month
Ramadan is the Holy Month in which God sent down the Qur'an. It commemorates the month-long fasting for Muslims worldwide, fulfilling one of the "Five Pillars of Islam".
God says in the Holy Qur'an:
"Ha Mim. By the Book which makes things clear. We sent it down on a blessed night; We are constantly giving warning." [Qur'an, 44: 1-3]
حم وَالكِتابِ المُبينِإِنّا أَنزَلناهُ في لَيلَةٍ مُبارَكَةٍ ۚ إِنّا كُنّا مُنذِرينَ
It is a month of extraordinary blessings. The Prophet ﷺ commented on this, as related by Anas bin Malik:
"There has come to you the month of Ramadan in which God has enjoined you to fast. In this month the Gates of Heaven are opened and the Gates of Hell are closed. Every devil is chained up. In this month God has ordained a night which is better than a thousand months of worship: whoever is deprived of its goodness is indeed deprived." [Ahmad Shakir, Musnad of Imam Ahmad]
قَد جاءَكُم رمضانُ شَهْرٌ مبارَكٌ افترَضَ اللَّهُ عليكُم صيامَهُ تُفتَحُ فيهِ أبوابُ الجنَّةِ ويُغلَقُ فيهِ أبوابُ الجحيمِ وتُغلُّ فيهِ الشَّياطينُ فيهِ ليلةٌ خيرٌ من ألفِ شَهْرٍ مَن حُرِمَ خيرَها فقَد حُرِمَ
Can I Say "Ramadan Mubarak"?
During the month of Ramadan, you will find Muslims greeting each other with a variety of greetings, most common being: Ramadan mubarak. But is there presedence for this in our religion? What are the allowances for cultural customs? Shaykh Joe Bradford gives us some proofs for consideration:
When discussing issues pertaining to custom and common practice, it is important to remember that the base ruling for all customs is that they are permissible as long as they do not contradict an established prohibition or contain a hazard to ones religion.
In the context of our discussion, not only is there no established prohibition for this type of greeting, but it is also void of any harm. Thus such a greeting in fact contains benefit, being that it is a manner of supplication by Muslims for each other, as well as a way to unite thehearts. Therefore this type of greeting is permissible. 
In addition to this general principle, several scholars of the past have mentioned narrations which support this issue specifically, such as the hadeeth narrated by Imam Ahmad in his Musnad:
Abu Hurayrah, may Allah be pleased with him, narrated that the Messenger of Allah ﷺ said while giving glad tidings to his Companions,
“The month of Ramadan has come to you, a blessed month; Allah has obligated you to fast in it, in it the gates of paradise have been opened and the gates of hell have been closed, and in it the Shayaateen have been shackled; in it is a night better than one thousand months, whoever was prevented its good has truly been prevented.” [Musnad Ahmad 2/230, 385, 425]
Commenting on this hadeeth, Ibn Rajab Al Hanbali said:
“Some scholars have said ‘This hadeeth is a fundamental evidence for [the permissibility of] people greeting each other with [the commencement] of Ramadan…So how can a believer not be given glad tidings when the doors of paradise have been opened? How can the sinner not be given glad tidings upon the closing of the gates of Hell? How can an intelligent person not be given the glad tidings of a time in which the Shayaateen are shackled…?” [Ibn Rajab, Lata'if al-Ma’arif, pg.281]
For further reading, I recommend the full article on Shaykh Joe's site.