How do photons carry color?

A photon physically has only a frequency, or a wavelength.  The speed of light is the product of frequency and wavelength.  "Color" is our brains' translation of the wavelength.  We see 400 nanometers as violet, 500 as blue, 600 as yellow, 700 as red.  So orange is around 650, green is around 550.  The wavelengths shorter than (let's say) 380 nm or longer than 720 nm are not visible to our eyes.  They are "ultraviolet" and "infrared" -- we can make use of their energy but we can't see them.

Color isn't a property of light. The color is determined by the response of the eye to different wavelength of the light. The center of red is about 600 nanometers and violet is about 400 nanometers. 

Color doesn't technically exist- its how our eyes interpret wavelengths. 

"Qualia" may be of interest to you.

They don't. They're an EM packet.

It's what our eyes do with those photons is what we perceive as color.

Minor correction to other answers: photons (light) carry color based on frequency, not wavelength. Proof: hold a green ball poolside. Then bring it underwater -- it's still green. The frequency is unchanged; the wavelength has shortened due to water's higher index of refraction (compared to air). Hence, color depends on frequency. The human body is tuned to see frequencies between 400 THz and 790 THz (terahertz).

Colour is how our eyes perceive photons that have different energies/wavelengths.  Each photon carries one specific energy and that energy triggers a receptor in our eye which indicates its colour.  It frequently triggers two receptors to varying extents which gives an intermediate colour between the two sets of receptors.  ie a photon can trigger red and green receptors. depending on the ratio of these it then is seen as red, orange, yellow to green.

Color is the wavelength of the photons. 

I don't think there is a definitive answer on this. Color is said to be a function of wave length. But light is not actually a wave, it is a packet of energy with a very small amount of matter. It is more of a pulse. Snell's law is based on how light refracts and there are what are called Snell's constants which are used to explain it. For some reason those constants also work with refracting sound and the law written that uses velocity to determine angle of refraction. Which suggests that the velocity of light might not be a constant. It might be that energy traveling inside of a set value of speeds (say +/- 10% of the speed of light) is detected as light. More likely there is a set of velocities that energy has to jump to (from the speed of sound) to travel. If it is less then that value, it is absorbed, if more then it becomes something else.

Also we may not have detection devices geared towards detecting energy traveling at speeds greater then the speed of sound but less then the speed of light. It is clear we have no detection methods now to detect forms of energy that propagate faster then the speed of light, but theoretically should exist.

The color of light might also be based on the exact rotation/s of the particle. Consider how a ball might rotate. The axis might be fixed and the particle spins on that axis. But the axis might also rotate as the particle spins. The combinations are almost (if not ) unlimited. When the particle is absorbed, it imparts a spin/s to a sub-atomic particle that our eyes register as a specific color because of a sub-atomic valance shift. Particle A was spinning in pattern S and is now in pattern Q so the color was R.

We don't understand so much about how the quantum computers (for want of a better term) are already in use by biological beings at least.