Is it wrong to say that an electron can be a wave? In QM it is sometimes said that electrons are not waves but they behave like waves or that waves are a property of electrons. Perhaps it is better to speak of a wave function representing a particular quantum state. 
But in the slit experiment it is obvious to see that electrons really are a (interfered) wave. So can you say that an electron is a wave? And is that valid for other particles, like photons? Or is it wrong to say an electron is a wave because it can be also a particle, and because something can't be both (a behaviour and a property)?
 A: What is a wave? From sound and water waves we come to an association with sine and cosine variational behavior. Wave equations are differential equations whose elementary solutions are sinusoidal .
In water waves and sound waves and even electromagnetic waves  what is "waving", i.e. has a sinusoidal variation with time and space, is the energy of the wave, represented by its amplitude.
When dimensions become very small, compatible with h, the Planck constant the individual "particles" electrons etc., can be described sometimes like classical billiard balls, and at the same time they exhibit a randomness, which when accumulated displays interferences and other wave characteristics.

This single electron at a time double slit experiment shows both effects. The individual electrons leave a point on the screen which seems random. The accumulation gives a probability distribution that has sinusoidal variations.
One can only give a probability for the electron to appear at the (x,y) of the screen, which depends on the quantum mechanical solution of the boundary value problem "electron scattering from two slits"
So it is not a classical particle  behavior because even though the energy is carried by the single electron, its (x,y) is  controlled by a probability distribution; and it is not the classical wave, i.e. a single electron that is "waving" its mass all over the screen interference pattern.  Each electron  is a quantum mechanical entity.
A: Electrons are neither particles nor waves - they are electrons.
We say they behave as particles or waves because we are familiar with macroscopic objects having these properties and want to provide a kind of "feel" for what they are in terms we can easily understand. We are the ones that select the experiment that shows aspects of their behaviour. They do not change from particle to wave and back again. Our experiments change.
A: In the micro world  particles  like electron has dual  nature .In some experiments  it behaves  like waves such as diffraction of electrons  by a single slit but in other experiments like  compton scattering or photoelectricity  it behaves like particle. 
In wavelike representation  of electrons by a quantum  mechanical wave function can explain the diffraction and interference of electrons.
A: Yes. No! Both! Neither?
The electron is an excitation of the QED quantum field, which is not quite compatible with the classical notion of either fields or particles. All you can do is draw analogies to either of these. Both analogies are sometimes just wrong, as in, they suggest different behaviour from what electrons actually do in experiments. However, they also predict some behaviour that agrees with experiment. And in the end that's all physics is about: finding models/analogies which allow you to predict the outcome of some experiments.
All of these models are wrong in a sense, but that doesn't mean you should never use them: just be aware that there are limits beyond which you get nonsense. It clearly is useful to think about the electron as a particle when you're designing a cathode ray tube. It is not really useful to think of it as a particle when you're trying to understand the spectra of atoms†... OTOH, a wave description works quite nicely here!
However, it is a sensible standpoint to say the electron will never be a wave, only its probability. Or perhaps better: (a particular kind of -)charge is a wave, but is quantised to something particle-like called electrons.
I rather like Dirk Bruere's approach: an electron is an electron, full stop.

†Even here there's a wrong but useful model.
A: The rules of how electrons move are analogous to waves because an internal state is cyclic and different possible paths are summed showing an interference pattern.
That's not the same as saying that electrons themselves are waves. The formulas for waves are used to explain where to find an electron.
A: The electron can be whatever we want it to be, we create the mathematical models in which the electron is a part of, so we can dictate the rules under which the electrons behave;
Quantum Mechanics is just a mathematical model, that includes mathematical objects like, "the electron" that we use to make predictions about the physical world. Just like we use numbers and fields, abstract objects like "electrons" are purely mathematical in nature, and we don't necessarily examine whether or not they "exist", because "existence" is a purely philosophical subject that's outside of the scope of the practice of physics.
In this way, we are free to dream of objects that can exhibit all kinds of "weird" properties, like the ones that particles have in Quantum Mechanics, that don't necessarily "exist" in a philosophical sense, but as long as our models work, we can keep on adding more and more layers of abstractions, because its our model, and we can make it whatever we want it to be! As long as it can make accurate predictions about the real world;
When we say that Gravity is a force, we don't necessarily mean that there is some kind of "rope" attached to the Earth and the Sun system, or when we say that Gravity is the curvature of the fabric of spacetime, we don't necessarily mean that there is some physical fabric that bends so that objects can fall into it, and create what we see as Gravity, either. Its all metaphorical and mathematical in nature, just a way for us to describe operations that will yield accurate predictions, in a yet consistent and rigorous mathematical way.
In that framework, the electron - we have made it - to be a propability cloud that extends EVERYWHERE in the universe. There is a small probability that we can measure the electron to be very far away (when we do measure it and its wave function collapses), but that propability is extremely low so it never happens...
So the electron IS its wavefunction, and that is not weird at all, because the electron is an abstract mathematical object so it can behave however it likes! Also for example, the electron can be a particle and a wave at the same time, because who cares? Its our model and we can have the electron be whatever we want it to be...
Will such a theory make accurate predictions about the Universe? If no, then we don't use it, if yes, then at the very least, we have proved that our theory is at least usefull.
So the only thing that we can say about the nature of electrons, is that in the framework of their models, they are helpfull mathematical objects, that constitute very helpfull theories, like Quantum Mechanics...
Answering "why" these theories are so helpfull, is outside of the scope of physics.
A: Dual split experiment showed e- can show wave like properties. When electrons were fired from a gun at a barrier with two slits on it the electrons exhibited a wave like pattern on the EM sensor behind the barrier, showing three distinct bands. Showing refraction occurred and hence electrons behaved like waves.  
