Some good answers so far, but let me just offer another way of looking at it.
You could say, as @amanderveen suggests, that while wavelength is an objective property of light, color is something that happens in the eye and brain. Taking that even further, you could say that we owe the existence of color to variations of a protein called opsin.
The receptor cells in the retina fire a signal whenever they are excited by light. At the heart of this reaction is a complex molecule called a visual pigment, consisting of the light-absorbing molecule retinal bound to opsin.
At the risk of oversimplifying, the retinal is what reacts to the light, and the opsin acts as the gatekeeper between the light and the retinal. If all of the eye’s receptor cells had the same combination of retinal and opsin, there would be no such thing as color.
What saves us from this monochromatic fate is the fact that opsin can combine with retinal in several different ways. Depending on the particular form of opsin surrounding the retinal, the retinal will only react to light in a particular range of wavelengths. If an animal’s eye has more than one of these possible combinations, the animal will experience some form of color.
Primates (including us) evolved three different visual pigments, one sensitive to wavelengths in the “red” range, one sensitive to the “green” range, and one sensitive to the “blue” range. People who’s eyes contain all of these have normal color vision; those with only two of these pigments are “color blind” (which really just means that they have a limited experience of color).
Other species evolved different pigments, and so would experience color differently from primates. Several species have more varieties of pigments as well, so they would see “dimensions” of color inaccessible to us.
The other side of color perception is, of course, the brain, which weaves the input from these color-specific cells into our familiar visual landscape. I know nothing of that piece of the process, so I’ll leave it for one of our resident neuroscientists to tackle.