It comes from the Jap word for "budding" plants, and is a term used to describe prepubescence girls going into puberty (yeah, it's weird), but also used to describe and Otaku's budding love for the kawaii character who captured their heart!!1!
It is most often used to describe characters acting "cute", such as being endearingly clumsy, ditsy, oblivious, or whatever "cute" qualities girls exhibit. In this case, it is closely related to the term "Kawaii", which literally means cute or small in Japanese.
The term is generally used to describe a cute character a person sees in an anime by an Otaku online, or describing a show as moe. Japanese men and girls may say it to another girl who is at the time acting "moe", You'll (generally) never hear this term applied to a guy. You will never hear it outside of Japan IRL, because it is often replaced by Weeboos with "Kawaii".
Additionally, one who obsesses over moe is often referred to as a moebro.
Despite /a/ getting fed up with moebros congesting their sanctuary to whine about anime dying and their pathetic lives, the epidemic just wouldn’t stop. One day, somewhere between 2010 in the depths of 2chan, a manifestation of every single moe character with their conspicuous physical characteristic, all into one moeblob, came into existence. There it was decided she would be named Moetron. Her reason for existence – to be hated and be made into thousands of guro rule 34.
Most Moetron threads consisted of just a picture of Moetron, thus creating a shit storm. Like the hideous creation of Frankenstein, this science experiment was hated by much of /a/, though it was also favored. Even the usual whiners of moe, being tsundere as always, came to like her.
- That K-On! character is moe!!!!~
- Oh, you try so hard! So moe!
- So moe I died (´･ω･`)