Paramecia
The most common of the three classes, users of Paramecia Devil Fruits gains superhuman physical abilities or powers. Other users can alter features of their bodies or their environment. Finally, there are some users than can manipulate and generate some kinds of substances.
Zoan
The users of Zoan class Devil Fruits gains the ability to transform into an animal (and acquire every ability affiliated with said animal). The user can also transform into a human-beast hybrid form. There are Zoan Devil Fruits not only for common animals, but for prehistoric or ancient animals and mythological creatures as well (ancient and mythological are more rare, with mythical said to be rarer than some Logia Devil Fruits).
Logia
The rarest of the three classes, users of Logia Devil Fruits gain the ability to transform themselves into an element (making them intangible or indestructible, the exception being the Darkness Logia power which keeps the human body and makes the user more vulnerable towards pain). They also gain every ability related to that element and gain the skill to disperse, control and rejoin any part of their bodies while conscious and in control of their powers (including explosions, bullet wounds or cuts). Logia users can move every part of their transformed bodies while in element form.