A convert is a non-Jew who saw fit to turn their back on the lifestyle in which they were raised in order to join the Jewish people. Throughout the Torah, God exhorts the nation to be careful not to mistreat converts. He is especially protective of them, since native-born Jews have fathers and big brothers and other family members to stick up for them. Converts frequently lack that kind of support system within the community, so He fills that role personally.
There is more than one prohibition against antagonizing converts. This one refers specifically to wronging them with words. The Sifra on parshas Kedoshim gives an example of verbal oppression. For example, it’s saying to a convert, “Yesterday you were worshipping idols…” (In our parlance, it might be saying, “Look who’s so frum all of a sudden!”)
The basis of this mitzvah is to get over ourselves. We’re no better for being born Jewish and we must be careful not to condescend to Jews-by-choice. God really drives this point home, as He reminds us about this in the Torah at least 36 times (see Talmud Baba Metzia 59b).
The prohibition against oppressing a convert applies to both men and women, in all times and places. In the Talmud, it is discussed in tractate Baba Metzia, pages 58b-59b. It is codified in the Shulchan Aruch in Choshen Mishpat 228. It is #252 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #49 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.