Moe is when your heart gets a boner.
Moe is a personal feeling, thus you could find whatever you want moe. Generally, it describes an infatuation with a girlish kind of cuteness however. The girl doesn't necessarily have to be a child however, it can also apply to a teenager or rarely even an adult, e.g. the deredere moments of a tsundere character for example.
The word cute captures an aspect of it, but cute doesn't necessarily imply moe, while moe almost always implies cute.
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