As with the other Passover-related mitzvos, the reason is to publicize the miracles that God wrought when He extracted the Jews from Egypt. Chometz is a leavened product of one of five grains: wheat, barley, rye oats and spelt.
The Zohar (2:182) equates eating chometz on Pesach to idolatry. Indeed, the extent to which we eschew chometz on Peach – not possessing it, not benefitting from it, destroying it, etc. – are comparable to the ways we treat objects of idol worship. The Rambam in Moreh Nevuchim (Guide for the Perplexed) discusses that seor (leaven) was not permitted on the altar because such was an idolatrous practice.
This prohibition applies to both men and women, at all times and in all places. It is #197 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #4 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar. It is discussed in tractate Pesachim on pages 28 and 35, as well as at the beginning of Kerisos. In the Shulchan Aruch, it can be found in Orach Chaim 443.