Full grain leather is leather that has not been split. It is the full hide. Not only has it not been split, but it has not been sanded, buffed, or any other alterations to remove any natural marks of the hide or imperfections on the surface. This type of leather retains the natural grain and skin pattern of the animal. Full grain has some breathability due to the natural pores. It also has natural oils, making it a very soft leather. During prolonged contact, there is less moisture on this kind of leather. Full grain also develops a patina during its lifetime. Some people like a patina on their jacket, while others take steps to prevent it from accumulating. A patina on leather is a gloss or sheen on a surface that results from aging. It is the highest quality leather out of all of them, and if you are buying a product that is full grain leather, the seller will be sure to make it known that it is full-grain. Full grain leather is likely more expensive, as it is higher quality and will last you much longer. For a leather enthusiast, or if you have the budget to afford the higher quality leather, full-grain leather may be what you are looking for.
Corrected leather is leather that has been sanded down to remove any imperfections. As a result of the sanding, the original grain is gone. A faux animal skin grain is then put on the leather by a mechanical pressing, and the leather is treated. Corrected leather gets treated with a few different products, including oils, treatments, and dye. This makes it more appealing and customizable to consumers, and results in a uniform grain with extremely smooth leather, albeit a more plastic feel. Corrected leather is always made from top grain leather, but not all top grain leather is corrected.
Genuine leather is the lowest quality leather. When full-grain leather is split, genuine leather is the bottom half, while the top half becomes top grain. Genuine leather is the catchall term for anything that is leather, but not high quality. It essentially just means it is technically leather, and can even mean scraps of leather bonded together to make the product (bag, jacket, belt, etc.)