Probability wave intuition Basically I am really new with this term. It all came before me when I was reading the Standard Double-Slit experiment.
An excerpt from Principles of Physics (by Resnick,Halliday,Walker) :

. . .by probability wave, to every point in the light wave, we can attach a numerical probability per unit time interval that a photon can be detected in any small volume centered on a point.

Now, this is not a theoretical definition; just an operational one and is applicable when the wave is about to interact with a matter. 
By knowing the intensity, we can find the number of photons associated with the wave. Then is the probability wave saying that we cannot specify the specific location of photons in the wave? But photons do appear only at the time of interaction,right? What does it , then want to convey actually?? What is actually the meaning of probability wave intuitively? 
 A: I think the answer you are looking for is from "Probability Amplitude" on Wikipedia:

Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic double-slit experiment, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that P(through either slit) = P(through first slit) + P(through second slit), where P(event) is the probability of that event.

There is also a video that shows a probability wave forming as a mapping of a particle's motions over time, from a theory known as "Pilot-wave".  For reference, this theory is still being studied, and is not accepted as the final answer to the wave-particle duality.  However, it does an impressive job of visualizing the how a particle, or set of particles, move in patterns that appear to mimic a probability wave.
A: Before physics discovered atoms and elementary particles (including photons) one could state that physics was the study of experiments and observations, i.e. measurements of variables with their errors, and modeling the observations mathematically. That is the way we got classical mechanics, classical electrodynamics and thermodynamics. The theories were formulated as second degree differential equations and the variables ,( position , momenta and energies for mechanics) were theoretically predictable and experimentall measurements only limited by the accuracy of the technology.
The solutions of second degree classical equations often have sines and cosines , and these described well waves in water and pressure waves and electromagnetic waves.
Then black body radiation, atomic spectra and the table of nuclear elements demonstrated experimentally the existence in the micro dimensions of structures that would not obey the classical theories. Energy appeared in packets ( spectra of atoms) and there were complicated rules of how protons and neutrons bound themselves into nuclei.
To start with the spectra in atoms were explained by the Bohr (planetary) model, a classical solution. In this model the electron around the proton forming the hydrogen atom and giving the Balmer (and other) series spectrum, a fit to the experimental observations, had to be constrained to fixed orbits by hand/postulate, and transitions from one orbit to the other  released the observed light/photon spectrum. (by then the photoelectric effect had convinced that light was composed of photons).
This was unsatisfactory because even though stable orbits could be calculated around a charge, there was no classical reason why the electron becoming unstable would not fall on the proton and disappear. Hence the need for postulates fixing the orbits.
Then came Schrodinger's equation and  solution for the hydrogen atom that reproduced the observed experimental series but also gave a general framework for describing data in the microcosm.
It is a partial differential equation and its solutions have sines and cosines and that is why they are called wavefunctions. It was necessary though to postulate several unorthodox ( for classical physics) operations and interpretations: the value of the observable variables could be predicted only by operating on the wavefunction with the appropriate differential operator and taking the integral over the phase space, the expectation value.  For the position x,

for the momentum, the operator is more complicated:

Because the solutions of the Schrodinger equation were often in sinusoidal form, the wave pattern could appear and interference phenomena as with classical electrodynamics.
Thus the quantum mechanical theoretical framework made sense when psi was interpreted as a probability distribution, and this interpretation is known as the Born rule.
The probabilistic interpretation of quantum mechanics has been validated by innumerable experiments and observations .  To develop an intuitive understanding can only come by studying and working on quantum dynamical systems. 
A: The wave is a description of the state of the photon. In quantum mechanics, every system exists in multiple instances that can interact with one another under some circumstances. In an interference experiment, you let those instances spread out so that the wave has non-negligible amplitude for multiple distinguishable instances and then bring those multiple instances back together. The probability of each outcome then depends on what happened to each instance.
For example, in a Mach-Zehnder interferometer, the interference pattern depends on whether you phase shift in either arm of the interferometer. If there was only one instance of each photon, then the probability of the photon being reflected at the first beamsplitter would be 1/2 and the probability of it being reflected at the second beamsplitter would be 1/2. So the probability of both reflections would be 1/4. The probability of it being reflected at the second beamsplitter after being transmitted at the first is also 1/4. So then the total probability should be 1/2. Instead the probability is conditional on what happens in each arm. The wave describes what happens to each instance involved in the experiment.
What happens when the photon is detected is that the detector differentiates into multiple instances,each of which detects one instance of the photon. When you look at the detector, you differentiate and so on for other systems. You can't interact with the other instances of yourself or other macroscopic systems as a result of a process called decoherence: 
http://arxiv.org/abs/quant-ph/0306072.
This description is commonly called the many worlds interpretation of quantum mechanics and is treated as controversial, but it is just a result of consistently applying quantum mechanics to all physical systems.
Probabilities can be used to describe this as a result of the limitations quantum physics places on how probabilities can be assigned to each instance
http://arxiv.org/abs/quant-ph/0405161
http://arxiv.org/abs/quant-ph/9906015.
See also "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch.
