When a non-jew converts to become Jewish, are they becoming Jewish racially then, as well as culturally or spiritually? Or is 'race' in relation to the Jewish people different to that of others, like black and white, etc.?

The concept of "race" is not totally accurate regarding the Nation of Israel=Judaism (as we explained in the past, one can convert in but can't convert out), and that's why we term it a "nation", rather than a "race" (also, we're not just a religion, for non-religious Jews are just as Jewish, as opposed to non-religious Christians who simply are no longer Christians). There are brown, black and white Jews, and there's no difference between them. It's true that our sages write in the Midrash 1,800 years ago, that Jews are brownish, but converts to Judaism since then have "expanded" the genetic pool, and it really never made a difference, and we couldn't care less what color a person is (e.g. in Israel there are several million darkish sefaradic Jews, several million whitish ashkenazic Jews, and over 100,000 black Jewish olim from Ethiopia). Inevitably, today, there is often marriage between these various groups, which we view as very positive and good for our national unity and moral values.