Abstracted from a Google+ discussion:
Technically, race refers to physical characteristics only. When people talk about race, they usually (pedantically speaking) mean ethnicity. Ethnicity includes physical racial characteristics, but also incorporates culture, nationality, language, religion, and so on. But if you’re not being a pedantic ass, you’ll use ‘race’ in its everyday sense as meaning ethnicity.
With that out of the way, here’s the key part:
What defines you as white (or black or Asian or…) is whether you can pass as that thing.
That’s all there is to it. As I’ve written before, there’s no genetic way to determine whether someone is black or white, because it’s not a scientific and genetically determined thing.
Race, as we deal with it every day, is a matter of consensus, so you are what you can pass for. It’s as simple (and as unfortunate) as that. The passing may require effort, or it may be effortless; it may be a choice, or it may be imposed on you. And as discussed before it may be contextually dependent.