I know that the light is reflected from a object to my eyes, but I don't understand exactly how. The photons appear from the light source and disappear in my eye! Can someone explain the phenomenon of where the photons go and do to allow us to see?
From the wiki article on color vision as an illustration of how photons are absorbed:
Perception of color begins with specialized retinal cells containing pigments with different spectral sensitivities, known as cone cells. In humans, there are three types of cones sensitive to three different spectra, resulting in trichromatic color vision.
Each individual cone contains pigments composed of opsin apoprotein, which is covalently linked to either 11-cis-hydroretinal or more rarely 11-cis-dehydroretinal.
So it is molecules with different absorption spectra that absorb the optical photons and start the sequence of giving a signal to the brain. It is not a simple matter and belongs more to biology than to physics. The physics part is just that the photon hits a molecule and raises an electron to a higher level which generates a series of reactions that finally register in the brain.
Photons can be created and destroyed freely, since they don't have charge or mass. Turn on a light, and you create many photons. Any body (made of atoms) not at absolute zero temperature will spontaneously emit photons.
They are consumed just as easily. Most any bit of bulk matter will absorb a photon in the electrons on the surface, transforming the energy into density vibration. No mystery; electrons (being charged) can do that. It's the opposite process of emitting via thermal vibration.
So where does it go… think of the electromagnetic wave, not the quantization of it. Vibrations in E field make electrons slosh back and forth. Moving charged particles create electric field changes in turn, which cancel out the wave and prevent it from propagating farther. Where does an ocean wave go when it hits the beach? It stops propagating so the wave (a phenomenon not an instance of an object) ceases to be.
The idea of particles makes you envision a thing that exists as an object, and that is misleading and detracts from the concept. The particle-ness in this case is just part of the rules that states that some physical interaction takes or gives energy on an all-or-nothing basis. That's seen in cases where an electron changes orbitals including when that is part of a chemical process. Bulk pigments that can absorb any frequency (in a range) freely still take exactly one wave's worth of energy at a time in units of amplitude described by Planck's constant.
Vibration — dynamics that can be started and stopped — is the underlying creation and destruction. Creating or destroying fixed sized units only is manifest in the rules for doing so, and does not represent an object in the sense that's bothering you.
More generally you wonder how something can disappear. Well, why not? Some stuff is conserved and can only move around; other stuff has no restriction. For producing light, you need to supply energy and balance the "spin". Those are individual attributes, not specific particles, and that is how such rules are generally found. An electron can be created if you also create a positron at the same time, to balance the charge total: you aren't destroying something or moving things from somewhere else, you are creating more things.
The current best model for explaining all this is Quantum Field Theory, where everything is fields and dynamic disturbances of them, with particles being emergent from the rules. IOW, just like the above explanations with the photon.
Where does a song go when you stop singing? It's a dynamical process, not a fixed object. It goes away when that process stops.
Light from all over the place hits your eyeball fairly randomly. The lens forces light from a specific angle to hit a specific part of the retina. This HowStuffWorks article shows how the mechanics of that work. The only major differences between camera lenses and eyeball lenses is that we can dynamically alter the shape of the lens to focus on different distances.
Now, your retina is composed of a bunch of rods and cones roughly arranged in a grid. They're a bit random, but you can think of them like your computer monitor: a bunch of pixels packed close together. For normal, color vision, the cones do most of the work, but the rods help too, especially in low-light conditions. Any time a photon enters the eye, it gets absorbed. Sometimes the cone absorbs it, turning the electromagnetic energy into electrochemical energy. Sometimes the photon passes through and absorbs into the back of the eye somewhere, turning the electromagnetic energy into thermal (heat) energy.
There are three kinds of cone, roughly corresponding to red, green, and blue. Red cones absorb most of the red light, but very little green light, and almost no blue light. Green cones absorb most of the green light, and blue cones mostly absorb blue light. If a few photons hit the cone over some time period, it sends a signal to the brain saying "dim light here". If a lot of photons hit over the same time period, it sends a "bright light here" signal.
The same thing happens for all the other cones. Between the lens that focuses light so one cone gets light from a small part of the world in front of you, and the several million cones in your eye, you basically have a giant Excel spreadsheet, encoded in an electrochemical format. By combining the information from multiple cones of different colors, the spreadsheet contains information about brightness and color at different angles in front of you. The opponent process is how the data is thought to be sent to the brain.
From this point, the brain does a bunch of black magic we barely understand using a bunch of seemingly-random code cooked up by millions of years of natural selection. It finds patterns in the data set and compares them to known patterns in both short and long term memory to establish what you're looking at, where it's at, what it's doing, who it is, etc. It also adds visual data to our spatial awareness centers, determines rates of motion, timing information, and probably a host of other things I don't know about. Then it ultimately sends this information to the rest of the brain to make decisions. Some of those decisions are made autonomously by low-level processing, while others are made at a high level with cognitive centers.
Imagine a spring-loaded trap with a hole that's sized such that only a particular size of object can enter the hole and trigger the trap. The molecules involved in vision are like that trap, with a bond having an electron energy gap tuned to the visible frequencies of light, encapsulated in a specialized protein that transforms the absorbed energy into a change in the shape of the molecule ('springing' the trap).
In short, the energy carried by the photon becomes kinetic energy to change the shape of the molecule.
After the trap is sprung, it must be reset by special enzymes that reconstruct the original shape. Note that 'setting the trap' takes energy (derived from your metabolism and hence the food you eat), which is what allows the vision system to amplify the relatively small energy from absorbing a photon into a signal that can be used to trigger transfer of information to your brain.
This is a gross simplification of the process, but those are the key points for where the energy goes and why such a small amount of incoming energy can result in such a complex cascade of processes.
Although there are already some excellent answers, I believe they are a little complex. Please allow me to offer a simplistic answer.
Let me start with the analogy of sound waves and the ear. The sound enters the ear and causes certain cilia to vibrate in response to the frequency and amplitude of the sound wave. Similarly a photon (as a wave), enters the eye and a given type of cone vibrates in response to the frequency and amplitude of the light wave. So, essentially, the photon is converted into an electro-chemical impulse that goes to the brain.