One may not use two different species, one kosher and one non-kosher, to pull a plow or a wagon together. We see that this prohibition is limited to a kosher/non-kosher combination based on the Torah’s example of an ox (which is a kosher animal) and a donkey (which isn’t). Biblically speaking, using a donkey and a horse (two non-kosher animals) to pull a wagon would not violate this prohibition. Neither would using an ox and a deer (two kosher animals) to pull a plow, though it would be a pretty weird thing to do. (Rabbinically, we may not plow using any two species that we are not permitted to mate.)
Man, though a mammal, is not considered an animal. If you wanted to pull a wagon alongside your ox or a plow along with your donkey, feel free.
The reason for this mitzvah is so as not to cause grief to the animals. From the ox’s point of view, the weaker donkey isn’t pulling his weight, which is unfair. Since an ox chews cud, from the donkey’s point of view, the ox gets to eat while it works, which is unfair. Basically, nobody’s happy with the arrangement! (The Sefer HaChinuch mentions a halachic principle that basically boils down to “birds of a feather flock together.” This concept applies to animals, as well! Additionally, the Rambam sees this mitzvah as a safeguard against keeping diverse animals in the same stall and coming to cross-breed them.)
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Makkos on page 21b. It is codified in the Shulchan Aruch in Yoreh Deah 297. This mitzvah is #218 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #180 of the 194 negative mitzvos that can be observed today as listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.