Particle motion can be like a wave, but how can we say that particle itself is a wave? I am finding it difficult to grasp the concept of waves itself.I am not talking about wavelength or frequency or amplitude.That can be easily visualized. They say water behaves like wave.But still water is still still without any waves! How can a single particle behave like a wave? Or is it the particle motion they are referring to?  Can anybody explain the concept of particles behaving like a wave, hypothetically using maybe a single sand particle?
 A: Quantum objects such as electrons are neither waves nor particles.
'Particle' and 'wave' are both ideas that we define using classical physics (not quantum physics) and then we can use the ideas as we see fit as an aid to understanding quantum physics.
A (classical) particle is a little bitty thing that can be located at one place at any given time, but may move from one place to another as time goes on.
A (classical) wave is a particular type of motion of an extended object. The parts of the object stay in one place on average, but as time goes on the parts move too and fro in a regular way. Ripples on the surface of an otherwise flat pool of water give a good example. The motion of the string of a stringed musical instrument gives another example.  Such motion can transfer energy and momentum from one place to another without any net transfer of the stuff that is supporting the wave (such as water in a pool or steel or plastic or catgut in a musical instrument).
I expect you can also get some good definitions from Wikipedia and other such resources.
When we do experiments with electrons, we find that some aspects of the behaviour are wave-like, and some are particle-like. The aspect that is on show depends on what the other physical things are, which the electron is interacting with. When incident on the surface of a perfect crystal, an electron with well-defined momentum will bounce off very much like a wave would. When incident on something like a photographic film an electron will instigate a chemical reaction at one spot on the film, much as a particle would. This is all very beautifully accounted for in quantum theory. The answer to the question, "so which is it: wave or particle" is "neither---but both concepts can be brought in as aids to the human imagination when we learn what the quantum theory is saying". 
A: When we talk about the wavelength of a given particle of certain mass, we find that its wavelength is given by the De Broglie wavelength equation given by λ = h/mv. The tricky part, like you asked, is trying to interpret what this means in a physical sense. Does it mean that the matter is actually a wave and has a wavelength given by that equation, or does it mean that the motion of the particle can be represented by that equation. The best answer that I, and probably physics in general has, is that this equation shows how we can represent the behavior of a particle with a certain mass and velocity. It means that particles can exhibit behaviors of waves, not necessarily that they are a wave, or that they move like a wave. Richard Feynman has a really famous lecture on wave particle duality that explains how this behavior is so non intuitive to anything else we understand physically, and these equations are just a way that we can represent their behavior mathematically. I highly recommend that lecture. 
A: The fact that a beam of photons or electrons can produce an interference pattern (with dimensions measured in centimeters) indicates that we are dealing with some kind of wave.  Probably each “particle” is associated with a wave “packet” of finite size. and with a fixed amount of energy and momentum.  (In an electromagnetic wave the energy can be ascribe to the energy density in the fields.)  The surprise is that all of that energy and momentum can be absorbed by an entity of atomic size or smaller. (As in the photoelectric effect.) This is what leads to the idea that there must be a “particle” associated with the wave packet.
A: A wave is some quantity that varies in a periodic way over time and space. For example, water-waves are periodic displacements of the surface level of the water; waves on a string are periodic displacements of the string from its resting position; sound waves are periodic changes in the ambient air pressure; light waves are periodic changes in electric and magnetic field strengths, gravity waves are periodic changes in the curvature of spacetime, and so on.
Waves have distinctive properties- namely interference, dispersion and diffraction. In classical physics only waves exhibit those effects. 
We used to think of electrons, protons etc as being particles, meaning some highly localised bits of matter, and light as being waves. However, we now know that in some respects light behaves like a collection of particles, and electrons, protons etc exhibit wavelike properties (eg interference, diffraction, and so on).
It's natural to ask the question about what is 'really' happening at  quantum level- are electrons really particles, or really waves, or really something else that mixes the qualities of particles and waves, and so on. The truth is that nobody yet knows for certain. Some physicists follow the view of David Bohm and others, who believed that electrons, protons etc are particles, but they are guided by some kind of 'pilot wave'. Others take the view that it is meaningless to ask the question about whether an electron is a particle or a wave. I think the best approach is to keep an open mind until new theories and experimental results shed more light on the question.
I think it is also a good idea to remember that quantum theory is just a model of reality, and as with all models it will probably be found to be an approximation of more subtle effects, so you shouldn't assume that the model will be an utterly authentic representation of what is 'really' happening.
A: Many answers and nobody is hitting the probability angle of quantum mechanics.
Classical waves, as sound, water, electromagnetic (defined as light) are transfer of energy in a space time dependence that can be modeled mathematically with a wave equation.
The simplest classical waves are  waves on a string :the constrained string molecules move up and down , and energy is transferred through the medium. One can find a number of videos explaining this.
There are second order differential equations, called wave equations , that model the wave behavior in classical physics.
It was found experimentally that even though light (electromagnetic radiation) can be described by Maxwell's wave equations, light needs no medium to propagate it, so just energy propagation through space can be well modeled with wave equations.
So in summary, the statistical behavior of energy transfer by many particles can be modeled with wave equations, as well as classical light which needs no medium to transfer energy.
Then came the puzzling data dependent on small dimensions, that could not be explained using classical theories: photoelectric effect, black body radiation, the spectra of atoms  are crucial because they needed a new theory for physics in the microcosm . This theory is called quantum mechanics. It was found experimentally that a differential equation , a wave equation could describe the spectra of atoms, and from then on the wave function  became the basis of modeling the behavior in the microcosm. 
The solutions of  the wave equation are picked up from the general mathematical set , by extra axioms, called postulates.
The postulate relevant for this discussion is the wave function postulate:
 $Ψ(x,t)$ = single valued probability amplitude at $(x,t)$

 $Ψ^*(x,t)Ψ(x,t)$ = the probability of finding the particle at $x$ at time $t$ 
                     provided the wave function is normalized

So, the wave at the level of quantum mechanics, is not an energy wave, but a probability wave. For a single particle it means that the probability of finding it at an (x,y,z,t) waves, i.e. follows a wave equations. This can be seen in this simple interference experiment, one particle at a time hitting two slits.

A single electron, a quantum mechanical particle, leaves a dot on the screen as expected form a classical particle, it is not spread out over the screen.But  the accumulation of electrons, going down on the images, shows an interference pattern typical of waves. The probability of an electron to hit an (x,y) on the screen follows a wave equation as the interference pattern demonstrates. 
This is where the concept of duality comes, which confuses people who do not follow the mathematical models. 
One has to keep in mind  that the duality is in the behavior of the particle when interacting . In the standard model of particle physics, the basic constituents of matter are considered point particles (which have no extension in space) and the model is very successful in describing a plethora of data and in predicting new .
