Photons have a property from the matter they interact with? When a photon leaves its source and hits our eye, our brain sees the source of the photon (Like a lightbulb or a star). When a photon is ejected from its source and bounces off of an object we see the object it bounced off. I hope 
I've been right so far.
My question is this: Do photons have some kind of property of the matter that they interact with that takes form when we observe the photon and see that matter? If so, where does that information go when a photon bounces off an object?
 A: A photon is an elementary particle with zero mass, moves always with the velocity of light c, and has energy given by E=h*nu . It has spin +1 or -1 and its wavefunction also has a polarization which will build up the polarization of the emergent from many photons classical electromagnetic wave.
Its energy is a property that characterizes the emitting particle/molecule/ion  at the source . If it comes from a specific atomic energy level it will have a definite energy . If it comes from plasma in a magnetic field ( as the corona of the sun) it will be in a continuum.
A photon can interact in various ways when hitting an object:
1) it can be absorbed completely raising an electron to a higher energy level. This may decay back to ground or cascade down, dividing the energy into more than one decay photons.
3) It may scatter elastically
4) it may Compton scatter off an electron or the field of molecules, changing energy
The information it carries because of its origin is lost when it interacts with the object. If a new photon comes out, a radiative decay of the excited level, it will carry that information in its energy until it is absorbed by the retina or the detector.
A: One single photon is not enough for you to see objects. If there would be just one photon coming to your eye, you would first see only darkness (before the photon arrives), then you would notice a dim point for very short time (when the photon hit your retina), followed by darkness again.
You can see objects because there is a constant flow of huge amount of photons in all directions. If you see a blue object, it just means that blue photons were bounced, while other photons were absorbed by the object. This already tells you something about the object (it reflects blue). In more complex cases you can even do spectral analysis of the reflected light and find the chemical composition of the object. So in a sense the reflected photons do carry information about what they bounce off. But you cannot make spectrum from one single photon. It must be a collection of photons of various energies.
Individual photons do not encode complex information about what they bounce off. You cannot say that some particular photon was reflected from wood, just based on that individual photon. Photons have energy (which is the color), they have momentum and polarization. None of these features can keep any complicated story about what the photon encountered. For example when you detect a blue photon, you can just say that it must have been born in some process that has energy equal to that of a blue light (2.5 eV), or it might have been Doppler shifted to that color. And you can say that the photon was definitely not absorbed between its birth and its detection (otherwise you would not be able to detect it).
The fact that you can see shapes of objects is due to the lens in your eye. The lens can "sort" the photons according to the direction they are coming from. All photons coming from a single location in the scene are sent to a single location on your retina. Photons from a different location in the scene are sent to a different location on your retina. The image of objects is created on your retina thanks to the fact that there is a lot of photons coming from every point in the scene into all directions, so your lens can fill the complete picture (all "pixels").
