Based on the link provided by Ataraxia and the Jisho entry, here's my attempt to answer my own question:
The general meaning of is something like "a state where all bad/unnecessary/superfluous/etc. things have been removed". Specifically, it can mean one of the following, depending on context:
1. feeling refreshed; feeling fine; feeling clear-headed - with respect to one's head, feelings, mood, etc.
>
> A cup of coffee cleared my head.
2. clearly; plainly; distinctly - e.g. sentence, article, writing style
>
> a clear writing style
3. shapely; neatly; refinedly - e.g. clothes, posture
>
> I'm looking for a coat. I'm short so the length should be on the short side, and as refined a design as possible.
4. completely; thoroughly - reaching a state where nothing remains
>
> Then clean it really well.
5. not at all (with negative sentence); not even slightly - e.g. "I don't know him at all". Basically, the opposite of #4.