Rotary tattoo machines and coil tattoo machines operate two different ways to achieve the same outcome of movingtattoo needles in and out of clients' skin to deposit ink into the dermis (secondary layer of tissue beneath the epidermis) with each insertion. It's what drives the needles that sets apart rotary and coil tattoo machines.
A traditional coil tattoo machine utilizes electromagnetic current passed through a pair of coils to trigger a draw and release of the machine's armature bar (shown to the left). It's a very hammer-like effect, with the release of the springs causing the armature bar to essentially tap the attached tattoo needles into the skin. Coil tattoo machines generate a buzzing sound that many people associate with tattoo shops.
Rotary tattoo machines, on the other hand, are incredibly quiet. There's a small motor encased in each rotary tattoo machine that moves the attached tattoo needles up and down in a smooth, almost cyclical pattern. Rotary tattoo machines move needles in and out of the skin more fluidly and evenly than coil tattoo machines do.
The two images below illustrate the motions of rotary and coil tattoo machines.* The image on the left is of a rotary tattoo machine. The cylinder with a red arrow around it is the tattoo machine's rotary motor; it rotates clockwise, moving the needle bar attached to it in a linear fashion, driving it up and down so that the attached tattoo needles will move in and out of the skin smoothly. The image on the right is of a coil tattoo machine, which uses a more complicated setup to drive tattoo needles in and out of the skin. When direct current from the tattoo machine's power supply reaches the two coils, they're charged and an electromagnetic field is created. That electromagnetic field pulls the tattoo machine's armature bar down (i.e. the part the blue arrow points to in the photo), which breaks the circuit created between the contact screw on the top-right and the metal front spring that was touching it, causing the electromagnetic field to collapse. A nanosecond later, the spring draws the armature bar back up again, placing the front spring back against the contact screw and re-establishing the electromagnetic field, which starts the cycle all over again. The breaking and re-establishing of the electromagnetic field is what makes a coil machine choppier than a rotary tattoo machine, which is powered consistently and therefore results in a smoother needle motion.
You can see all the parts of a coil tattoo machine in more detail in the graphic on the left below. The graphic on the right shows the components of a Cheyenne rotary tattoo machine; not all rotary tattoo machines have this same layout, but they include many of the same components and function in generally the same way.