What is importance of Ramadan in Islam? How to wish Ramadan Mubarak?

Ramadan is the name of the ninth month in the Islamic calendar.  Muslims are enjoined by the Qur'an to spend that entire month re-dedicating themselves to their faith by fasting and contemplation from sunup to sundown.  This is seen by many as preparation (or perhaps purification) for the year's mass pilgrimage to Mecca, (called "Hadj", which Muslims should make as often as they are able...although for most it is a once-in-a-lifetime trip).  Many also feel that God has revealed to them that they should make a point of celebrating and appreciating family and their marriages each night until the beginning of dawn, throughout Ramadan.

Because the Islamic year is lunar, the month of Ramadan doesn't fall in the same place among seasons every year.  (Nor does it stay aligned with any month in any non-lunar calendar.)  Instead, it slowly makes its way earlier and earlier into the seasons as the years go by.  As a result, sometimes Ramadan falls during comfortable seasons and in other years it falls during extreme heat.  That, combined with the challenge of neither eating nor drinking all day, plus the lack of sleep that some families inflict upon themselves in keeping with the admonishment to celebrate family until dawn, often makes Ramadan a difficult trial for many.

However, it is one of the major features of the faith's calendar - and Islam, being a faith which accepts Man's tendency to be distracted by worldly things, places emphasis on frequent reminders of the spiritual foundation of all Creation (which is why Muslims are called to pray five times each day, and enjoined to fast for a month every year).  Other devotions are prescribed in the Qur'an itself for those who are sick or who must travel, or who have other duties that make fasting ill-advised.

While it's possible to express best wishes at the start of Ramadan, that's not normally done by most Muslims.  That would be akin to a greeting card that said "Happy Lent".  Certainly some Christians might welcome Lent with excitement - but they tend to focus on the joys of the Easter season at its end, instead.  Likewise, most Muslims focus on the end of Ramadan and the holiday (or "eid") called "al-Fitr", which is usually celebrated generously with family, including gifts for the little ones.  So it's far more common to hear Muslims express "Eid Mubarrak!)