What causes atoms to have their specific colors? I understand that light (color) is part of the electromagnetic spectrum, and that it depends on what wavelengths are reflected/absorbed. Though what property of an individual atom gives it its color? Do electrons, neutrons, and other subatomic particles have color?
Basically: What intrinsic property causes the differences between how the varying wavelengths of light are reflected at the atomic scale? Also, how do photons factor into this?
 A: Color is a purely psychophysical creation of the human eye, in response to EM radiation in the single octave from 400 to 800nm wavelength (air), but most older people have negligible response from 700 to 800 nm.   "Light" is likewise a creation of the human eye.  That's why we use separate units of measurement for it; lumens, candelas etc.
Atoms etc have NO color.  They create EM radiation energy,at various wavelengths, only some of which evoke the human eye "light" response. Scientists need to watch their language, and use correct terminology, if they want to convey useful information, and not confuse students, with loose language.
"Light" by definition, IS visible.  It IS NOT   UV or IR.
A: 
Basically: What intrinsic property causes the differences between how the varying wavelengths of light are reflected at the atomic scale? Also, how do photons factor into this?

These are absorption lines in the solar spectrum


Fraunhofer lines coincide with characteristic emission lines identified in the spectra of heated elements.6 It was correctly deduced that dark lines in the solar spectrum are caused by absorption by chemical elements in the Solar atmosphere.

Emission lines of Iron are shown below.
If you have some physics background you will know that both these spectra depend on photon emission and absorption by the electronic states around atoms and molucules.

As there are many elements in the sun part of the observed continuity of the spectrum is due to the overlap of frequencies and the multiple possible states for each atom and molecule to absorb/emit  photons.
There exists also continuous emission of photons when electrons ( charged particles) are accelerated or decelerated in  external spill over electric and magnetic  fields that exist around all atoms and molecules. This spectrum, called black body, will be continuous, and is what one sees in an incandescent lamp or very hot iron; photons are emitted  continuously by all bodies , even though not in the visible spectrum. 
So the role of photons is crucial to all electromagnetic radiation, including that from our sun. As an ensemble they form the classical light wave, the creation process is a quantum mechanical one involving photons.
Reflection is again an interaction of the individual photons of the beam with the material they impact. The frequencies that  are not absorbed  will define the color of the material that scattered the beam.  So photons are crucial to defining the colors we see.
The photons are coherently scattered by the  field of the electrons of the atoms and molecules of the scatterer. The classical view of an electromagnetic wave can be shown to  coincide with the quantum mechanical one and is much easier in calculating the behavior of beams, which are huge ensembles of coherent photons.
A: What frequencies are reflected/absorbed depends upon the structure of the atom (which, as quantum mechanics has taught us, is something really hard to understand). So to know what particular property of the atom decides which frequency/wavelength it should absorb/reflect is very hard. 
As for electrons and other subatomic particles - they do not possess any color, so to speak, since visible light has a wavelength too huge to reflect off these particles. That is why you must use X-rays and electron microscopes to observe atoms (and we still can't see subatomic particles, though I really do wish we could - finally the whole uncertainty thing will have a whole new angle to it, don't you think?).
A: Color has little to do with discrete absorption spectrum. Sun light has that spectrum but it has no specific color. Sun light is white.
If a material absorbed wide range of continuous bandwidth of light, it would be showing the color of remaining bandwidth ( not absorbed part ). But this is the rare case. The colors of materials we have seen in daily life are not formed and working this way.
Color is formed by huge number of atoms, it's a property of substance or material, not property of a single atom. Similarly, water or other liquid has viscosity, a single water molecule doesn't.
A: First of all, color is not reflected. When light is reflected it creates glare which is an image of the light source itself.
Color is created either when (1) light of a particular wavelength is emitted from a light source or (2) is absorbed and then re-emitted by a physical object. So, when we say an apple is red for example, it refers to the second case.
All molecules can aborb electromagnetic energy, including energy from the visible spectrum. They then re-emit some of that energy as new light, but the spectrum of light they emit will be different than what was received. For example, a white light, containing energy from all parts of the spectrum, might be re-emitted by a red object as primarily red light because energy in the green and blue parts of the spectrum have been preferentially absorbed by the object.
The apparent color of an object depends on the strength of the energy emitted at each frequency. So an object that emits lower frequencies of visible light will appear red-yellow, and one that emits higher frequencies of visible light will appear bluish-purple.
