This word is most popular phrase in Japan now, and it is really hard to explain in English at the same time.
“Moe” is written as “萌” in Kanji (pronunced like Mo-eh). It originally means “sprout” or “bud”. However, “Moe” as a slung is very different.
This word was born in Akihabara, Tokyo which is famous as a town of electrical appliances. You can buy any kinds of electrical things there, even tiny parts of radio and minute parts.
There are many, what is called, “Otaku” who are crazy about figures, anime (cartoons), or pretty characters in a cartoon in Akihabara. When these people see pretty girls and feel happy, they express their feeling as saying “Moe”. They seem to feel relieved and relaxed when they see their idols or girls who wear costume of maid. It became now that not a word only for Otaku people but also for people in any different generations in Japan. Probably, men and young girls use the word when they find “Moe” girls.
Japanese anime, "Sailor Moon" http://hicbc.com/tv/sailormoon/ and "Evangelion" http://www.bandaigames.channel.or.jp/list/eva2/ are typical "Moe anime".
For example, "Sailor Moon is chou moe moe." (you can repeat moe to emphasize.)
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