Moe is a character design intended to induce an irrational desire to hug a character in the viewer. It has become increasingly common since the early Naughties and the success of Azumanga Daioh (though it's quite a bit older than that). Indeed, we might describe it as an attempt at intentionally recreating viewer reactions to Rei Ayanami (who many fans consider moe, much to Mr. Anno's chagrin).
Alternately, the concept can be summarized thusly:
- Moe! Moe! This is an element of moe. Fundamentally, in every story where something strange happens, there's always an alluring, lolita-looking character present. That's not all! She's so small, but look! Her breasts are bigger than mine! A lolita face with big breasts! This is an important element of moe! They are really big! It's starting to piss me off! Such a cute face is sporting bigger ones than me!
--Haruhi Suzumiya, epic troll, as she molests Mikuru and explains moe to Kyon
A moe series is a show that revolves primarily around moe characters. Some people find this annoying. Others find it refreshingly relaxing.
A moe blob is a character that has no purpose to the story other than to be cute, huggable, and adorable.