Seems like Lili's covered quite a bit, so I guess I'll just talk about (or expand on) elements and their associations with fairies and you can choose whether you want to use it or not.
As with many facets of fantasy, fairies often govern an element. The usual ones, such as fire, water, earth, and air. These are just a few, as they can also govern light, darkness, various emotions, so on and so forth, but unlike the True Fae, these fairies are born with their affiliation to the elements. Examples of such fairies include a Sprite, which acts as a water fairy, a Slyph, associated with the air (and as such are invisible and actually being made of it) and a Nymph, which is a type of fairy that is a deity of an element of nature and, in some cases, have the appearances of them (for instance, Dryad's often have the appearances of women, but various interpretations have them as (usually malevolent) trees). As such, just because a fairy governs an element doesn't mean they can use magic associated with it.
Now, fairies are magical in many interpretations, often with the ability to cast magic, though are usually weaker than other beings that can cast magic (including humans). This is largely because of their neutral natures. Fairies are, for the most part, not fighters, and they shouldn't be seen that way. Sure, they can be dangerous, but they aren't inherently made for fighting, nor are they inherently aggressive. Usually, they learn magic naturally if they can, often to protect themselves and whatever they govern.
Also, despite they're mostly mischievous nature (especially Pixies), they are not inherently "evil" (well, most of them, at least). Rather, it is simply in their nature to want to pull pranks, both big and small. "Pranks", in this case, is a relative term, because what a fairy may see as "a little fun mayhem" may be something as dire as destroying objects, destruction of a house, or in some cases, killing (though their intent is still neutral). Fun, right?
Though this is mostly general information, as a "Fairy" is a general term. There are TONS of interpretations and subclasses of fairies, so it's really up to you to choose which characteristics you want to use to culminate in to a single species called "Fairy".