what is the difference between Доброе and Добрый? Both mean the same but when we say good morning we use the former instead of the latter , as per this site http://masterrussian.com/blday_greeting.shtml
Well, it is just a grammatical gender indication.
Доброе утро is "доброе" because "утро" is neuter in Russian.
Добрый день and
добрый вечер contain
добрый since both "день" and "вечер" are masculine.
Check out this article for further information.
Still, there's one subtle issue I want you to be warned about.
While it is grammatically valid to say "добрая ночь" (good night), nobody says this in Russian. The valid form is de-facto
доброй ночи or
спокойной ночи and this, surprisingly, this is more about declension than about grammatical gender (but gender is important as well, has the noun been masculine, form "доброго" would be used). In this particular case, the adjective "добрый" is genitive case, because "доброй ночи" is actually is a shortened form of "[I wish you] a good night". And in Russian verb "wish" in this case is governed by genitive.
Just to add my two cents to the @shabunc's answer (which is itself good and full).
There can also be
доброго дня and (however rarely used, obviously because it's a rare case to say goodbye in the morning)
доброго утра. Which would mean: have a good day, have a good morning. The situation with good night when you can't greet anyone saying "добрая ночь" seems to happen because there was no "night life" as nowadays and thus no one would have needed this form of greeting. So that, all the languages have some traditions, Russian just doesn't have a tradition to greet at night.