This is a very popular question, It seems like many people are confusing waist trainers to corsets.
Both garments are used for the same purpose, to achieve a smaller waist.
However, they do it in a different way.
Waist Trainers are what I like to call the modern and safer way to waist train. These garments are made of latex and cotton mix or just latex depending on the style and brand.
The materials used on waist trainers allows them to stretch making it possible to waist train comfortably while sweating off belly fat, unlike a traditional corset.
The way a waist trainer works is by compressing your waist while the latex causes your waist to sweat out fat toxins and gradually shrinking the fat cells. The steel or plastic boning it has helps by keeping the waist trainer perfectly straight and not allowing it to roll up. This also helps you maintain a straight posture when waist training.
Corsets, on the other hand, are made of many different materials such as cotton, silk, denim, leather and many other materials mostly nonstretch.
They include anywhere from 6-24 steel or plastic boning and are designed in many styles and shapes.
The way a corset works is by pulling in your ribcage and torso into a desired shape. This practice is also called rib training. This training does not cause your waist to sweat out fat unlike a latex waist trainers can. A corset's purpose is to squeeze your waist into an hourglass shape with extreme compression.
A Corset can be a bit impractical to wear daily unlike a waist trainer which can be worn underneath your clothing discreetly.