Cone cells are one of the two kinds of photoreceptors in the eyes (the other being rods). Cones, as you infer in your question, are responsible for the perception of color. One thing that should be clarified is that people don't only have 3 cone cells, but rather have only 3 kinds of cone cells, and many of each kind within the eye.

The 3 different kinds of cone cells people typically have in their eyes are: S-cones, M-cones, and L-cones. S-cones absorb wavelengths of blue light, M-cones absorb green, and L-cones absorb red. The differences in the types of cone cells is the result of being comprised of different kinds of photopsins, or photoreceptor proteins that are typically receptors that undergo a conformational change when activated by light. So for the S-cone cells, blue light activates the photopsins in the cells, whereas the M-cone photopsins are activated by green light and the L-cone photopsins are activated by red light. Once these conformational changes take place in the protein, a series of signals are sent throughout the cone cell and eventually, all along the pathway to the visual cortex in the brain, which forms images based on the signals it received.

People who have 3 kinds of cone cells thus see in something akin to the "RGB" (red-green-blue) based colors we're used to seeing on a computer screen. People with 4 different kinds of cones, however, see in RGB + some variation/"shade" of R, G, or B. Ultimately this corresponds having an absorption peak at a slightly different wavelength that still might correspond to some "shade" of R, G, or B. But now, because of this extra responsiveness to a fourth "color," there is a larger number of combinations for the shades of colors perceived.