Yui-chan: 1000% of your daily recommended intake of moe, yet not even the most moe thing in the show.
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).
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.