I've seen people trying to define it in many places, but all of them end up giving different versions or saying it can't be defined.
2 Answers
There are multiple ways "Moe" or 「萌え」can be used.
「萌え萌え」as a mimetic word (擬態語) like Ignacio's link in the comment.
「萌え！」as an interjection (感嘆詞)
「妹萌え」as a prefix (接尾辞) In this example someone has a fetish/attraction/affection/情熱/欲望 towards little sisters.
「萌える」as a verb as well. For example, 「あなたは（私から見て）萌える」
Either way you can attach 萌え to almost anything that you might be attracted to in regard to females (I've yet to see it directed towards males, but if it exists please correct me). However, 萌え is generally used in the context of anime, manga, video games, etc, to denote someone's charm (which may or may not be recognized by a specific group of people).
What your asking about is the slang version, but originally 萌え comes from 芽生え which means to bud or sprout in terms of plants.
"Moe", from 「[萌]{も}える」, refers to the feelings of the subject towards the object, especially with regards to affection or care.
answered Jul 25, 2012 at 7:31
萌{も}えthe English Wikipedia article has a pretty good rundown on it, but it's got a lot of usages which may not be explained there.