While "chest protector" is fairly self-explanitory, the plastron is one of the most confusingly named pieces of equipment fencers use. In zoology, turtles have a part of their shell on the underside of the animal covering the chest area that is called the plastron. Historically, the plastron was a metal breastplate and in fashion it was a bib-like panel on dress or shirt.
In fencing, the plastron is a decentent of these other uses of the term, being a puncture resistant, fabric piece of protective gear. A plastron will have one short sleeve and a panel that covers the upper chest and underarm on the torso. A fencer will only need to wear a plastron on the arm that they use for fencing (the weapon arm). The plastron is worn under the jacket.
The reason for the plastron is to provide additional security to the saftey of the jacket, adding another layer of material designed to withstand direct hits. The seams on plastrons are made to offset the seams in the jacket, so that there should be failure in the jacket, you have another layer of defense.
The chest protector is a rigid, plastic chest piece that covers the entire front of the upper torso/breast area from the sternum to the clavicle. In competition, they are required to be covered in a compressable material to increase comfort and reduce "tip sliding" off of the hard surface. They are required for women fencing in competition and are optional for men's events. The chest protector is worn over the plastron and under the jacket.
These come in a basic range of S, M, L, etc sizes. They can be purchased for men or women, "mens" will be a flat piece curved to conform to the torso (think Storm Trooper armor), for "womens" are shaped to have space for breasts. Many women prefer the simplicity of the mens chest protectors, not wanting the "shaped" pieces with the artificial "anatomy".