Today is the start of the holy month for our Arab friends. This is their time of fasting, just like with us Christians during our Holy Week. Ramadan usually starts from 5am til 7pm and just like the start day, its not constant  every year. It depends on the timings of the moon for a certain year. 2010's start is early, as they say, coz last year it was September.
Ramadan kareem is a greeting, that means "Ramadan is generous", "Ramadan is good" or "Have a generous Ramadan". It is like saying Ramadan Mubarak which means "Have a blessed/rewarding Ramadan". Either of the two you can use to greet Arabs and friends during this time of the year.

Ramadan is a month to reflect, to pray, to fast on food and on earthly desires, which will be rewarded at the end of the day (later at night), where most of them eat voluptuously, cheer, and have great joy. And it would be unwise and unlawful to play with your food teasing those who fast while they are restraining themselves from food, so it's a no no. If you're not Muslim, and have an urge to eat, go to a place where no such can see you (tip like in a washroom = nightmare).
A month after is their EID, or what we term as New Year. It is spent for 3 days, where Arabs give monetary value to none Arabs, I say lucky those who receives hahahaha.