A2A
There are three categories of Devil fruits:

1. Zoan: The most common of the three, it gives its user the ability to turn into a animal/Human hybrid to various degree, depending on the control of the user. It has various subclasses based on various species of animals it turna the user into, with the super rare Mythical Zoan being extremely coveted. Zoan fruits can also be eaten by inanimate objects (I don't know how, ask Vegapunk).These can be artificially produced as well.
2. Paramecia:Change in body structure of consumers, giving them ability to change their environment or themselves, produce beams or growth. Examples of Paramecia fruits are:Nico Robins Hana Hana fruit which helps her grow hands anywhere, Boa Hancocks Mero Mero fruit which lets her turn people to stone using beams, Luffys Gomu Gomu fruit,which changes his body to rubber.
3. Logia:The rarest of the three, logia users have their body changed to natural elements. The distinction from Paramecia like Luffys is that the logia users can generate the elemnet their body is composed of freely. They become nearly invulnerable as they can just absorb any foreseen attack, though they are likely to get cocky for the same reason, also they are vulnerable to Haki attacks and can be caught off gaurd. There also exists an order of superiority among the elements. For example Aces Mira mira Fruit lost out to Sakazuki(Akainu) and his Magu Magu Fruit as His heat was stronger than Aces fire