What is moé? Well that depends on the subject. There are two definitions of the term, simultaneously distinct, and yet intertwined. The original definition, somewhere along the way evolved into something new, and in contemporary use, Moé now encompasses both meanings, often simultaneously. This article will first define the root meaning, and then the "current" definition.
Moé is derived from the Japanese word 萌える (moeru), meaning "to sprout" or "to blossom". Not to be confused with 燃える (moeru), meaning "to get fired up" or "to burn". With respect to its context, it can be used as both an adjective or a noun (or technically even a verb).
Moé is a feeling of platonic affection or fascination towards a certain target. On OT, it is almost always used in reference to female characters, but can technically apply to effectively anything at all. Even inanimate objects can be considered moé. It can refer to something as simple as wanting to hug a character, or a desire to protect him/her from the world. It's a pure feeling, removed from any sexual connotation. Mio Akiyama, a character from the anime "K-On!" is often considered a moé character because she is easily frightened. Her friend Ritsu will say something that Mio interprets as frightening, and begin to curl up into a ball with tears in her eyes. The feeling on the part of the viewer of wanting to go comfort Mio could be considered moé. Equally, a desire to stop Ritsu from picking on her would also fit the definition. Even the feeling that Mio is cute simply because she cries so easily could be described as moé. In the most general sense, it refers to a specific personality trait, and the feeling that trait produces.
Another good example of Moe is Yutaka from Lucky Star, who despite being a 1st year highschool student is as small and innocent as a loli; She is also on the sickly side. But as a result a high moe response is drawn from most people who want to protect her and her childlike innocence. (Perhaps not certain borderline pedos)However, it is not confined to personality traits alone. Physical features, clothing, personal history and even disabilities are definably moé. As a result, many common fetishized costumes fall under the definition of moé, which leadsinto the second definition. Rather than a personality trait that inspires platonic affection, in contemporary use, moé is more commonly used to refer to traits that are considered "turn-ons". Maid outfits, school swimsuits, cat ears, gym uniforms, and many other highly fetishized traits fall under the umbrella of moé. In practical use, the two definitions are often simultaneously applicable. A feeling of affection towards a character may begin with what is considered a desireable personality trait on the part of the viewer, but quickly change into a feeling of physical desire. Indeed, it's rare that a moé character would NOT fall under both uses of the term. e.g. "She's so cute and huggable, I'd feel bad tapping that...but I totally would anyway." (Most Curvy Moes might qualify in this regard)
While much less common on OT, moé can be applied to real-life targets as well. Generally it is applied towards popular idols or seiyuu. It can also be used in regard to animals or objects. Cats, for example, are often considered moé, and putting out-of-the-ordinary outfits or accessories on an animal can be said to increase its "moé factor". e.g. putting plush wings on a dog. (It should be noted however that this applies strictly to the traditional definition of moé. At least I'd like to think that OT-kun doesn't go in for that kind of thing...)