It is a mitzvah to love a convert to Judaism. This is despite that fact that a convert is already included in the general obligation to love every Jew (Leviticus 19:18). This is similar to the way that there are special mitzvos not to cheat a convert or to verbally abuse him (Mitzvos #64 and #63, respectively), despite the fact that we are not permitted to cheat or verbally abuse anyone. The reason we stated there is that God takes extra steps to protect a convert since they lack the support system of a Jewish family. A convert can't turn to a Jewish father or Jewish big brother for protection, so God personally fights the convert's battle.
To love a convert means to go the extra mile to avoid causing him pain. We should love them with an intensity approaching that reserved for God Himself! If a convert needs our help and we neglect his needs or if we disparage him because of his background, it is a serious violation. (We are likewise to help anyone who is away from home and cut off from his usual support system, though that is just a moral lesson derived from this mitzvah.)
The reason that God has commanded us to love one another is so that all Jews should be like one big family and that there should be peace. The convert is extra deserving of this consideration because he so desired the truth and to cling to God that he was willing to leave his family and his people to join the nation of Israel. Sadly, too many people do not give converts proper credit for their sacrifices.
The Torah tells us that we should love the converts because we were strangers in Egypt. We've been the outsider, oppressed and cut off from aid. Having been there, we should know how badly it hurts and do anything in our power to prevent others from feeling that sting.
This mitzvah applies to men and women in all times and places. It is discussed in the Talmud in tractate Baba Metzia (59b). This mitzvah is codified in the Mishneh Torah in Hilchos Deos chapter 6. It is #207 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos and #61 of the 77 positive mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.