This is pretty straightforward: don’t take revenge. That seems fairly clear. What may surprise us is the extent to which we are forbidden to take revenge. Rashi cites the Sifra on this verse that the definition of revenge is refusing to lend someone your shovel because he had refused to lend you his hoe. To us, such refusal may seem a perfectly reasonable course of action. The Torah, however, considers it a form of vengeance.
It would be easy to say that the reason for this mitzvah is to encourage peace among people, but the Sefer HaChinuch says something surprising. He attributes this mitzvah to helping us understand that God is in control. When bad things happen, who should we blame? It’s easy to say that the other guy is at fault, but it’s really God Who is in charge. If we’re upset and aggravated, maybe we did something to deserve being upset and aggravated. Rather than spiting the other guy, we should take a look inside ourselves and evaluate our own deeds.
This mitzvah applies to both men and women in all times and places. It is discussed in the Talmud in tractate Yoma on pages 23a and is codified in the Mishneh Torah in the sixth chapter of Hilchos Deos. This prohibition is #304 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #80 of the 194 negative mitzvos that can be fulfilled today as listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.