Can you please show me a final atomic model which demonstrates movement of electrons inside it? Is there any final model of an atom, of which we can say, “This is it”! Or is it still improving and physicists are not completely sure about it? I am particularly interested to know how exactly electrons move inside an atom. Do the physicists have any actual image or video of electron orbitals?  
 A: What we have is Quantum Mechanics  supplemented by Quantum Electrodynamics. With the tools available you can calculate atomic properties to increasing accuracy. Neutral hydrogen can be treated by the Schrödinger and more accurately the Dirac equation. Then you can throw in perturbative QED radiative corrections and a finite size nucleus. This brings you to the limit or beyond experimental accuracy. For many-electron atoms you also have to consider configuration interaction and corrections to the Born approximation. This is quite final in my opinion. 
A: Electrons do not move inside atoms.
If an electron is in a given energy level $E$, the wavefunction is given by $\psi(x,y) = \phi(x)_{n\ell m} \,\mathrm{e}^{-\mathrm{i}E t/\hbar}$. The time dependence is a pure phase factor, hence the real-space probability density of the electron is $|\psi(x)|^2 = |\phi(x)|^2 \neq f(t)$, not a function of time. These are called stationary states, for this reason.
The fact that electrons do not actually move in atoms is good, and it's the whole point quantum mechanics was invented. If they were to move, they would be accelerating charged particles and would thereby lose energy to radiation (bremsstrahlung) eventually collapsing. The instability of the atom was exactly the shortcoming of classical physics that led to the invention/discovery of quantum mechanics.
Furthermore:
Atomic orbitals are only "correct"$^\dagger$ wave functions in one-electron systems such as the hydrogen atom. In many-electron atoms, orbitals are a useful approximation, usually a basis used for perturbative calculations. For instance, for Helium you already have to take into account the indistinguishability of the two electrons, which leads to the linear combinations of the orbitals to work out correction terms.
In the Hydrogen atom, the orbitals have been indirectly observed, see Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States, by recording the diffraction pattern of light radiating away from atomic transitions: these patterns related to the nodal structure of the atomic wavefunctions.
Angular-resolved photoemission spectroscopy (APRES) can give information on the shape of molecular orbitals, see Exploring three-dimensional orbital imagingwith energy-dependent photoemission tomography. 
$\dagger$: but only within the pure Coulomb Hamiltonian. With corrections such as fine structure, Lamb shift etc., there is no analytical solution for both eigenvalues and eigenstates.
EDIT from comments.
Given the attention this answer has got, let me add a few points raised in the long discussion that ensued in the comments.
First and foremost, the above answer reflects my opinion and my interpretation of the matter. Indeed, as @my2cts points out:

Whether electrons move or not is pure interpretation. What QM does
  unequivocally say is that electrons have kinetic and potential energy.
  Anyone is free to interpret this.

Then, regarding motion, it is true that electrons possess momentum, kinetic energy, and, for $\ell \neq s$, a probability current $\mathbf{J}$ that is however also stationary but in the tangential direction $\hat{\boldsymbol{\phi}}$ (derivation here) like the velocity of a classically orbiting object.
Particularly, @dmckee says:

the electrons have a well defined energy which has to be interpreted
  as including a kinetic component and a momentum distribution which may
  include zero but also includes non-zero value with non-trivial
  probability density.

My idea of "electrons do not move" stems from the idea that "standing waves do not move", in that they don't go from A to B. But of course there is motion nonetheless. See nice discussion here.
A: It's impossible to have the final model of an atom. We'll have to do it with approximations.
Ther are simply too many factors to take into account. Even a "simple" thing as the proton spin: See this PDF for example.
And, depending on the kind of atom, there is at least 1 proton to find in every atom. And don't forget the neutrons (both nucleons). and the interaction between them. The spin-orbit couplings. Etc.etc.
For the proton-electron system (Hydrogen) the best approximation can be made. Simply because it's the most simple atom. Three quarks (proton), one electron. 
It is already difficult though (even with the aid of a supercomputer) to calculate the interactions between the valence quarks and the non-valence quarks by means of gluons (see the PDF).
When this is done you have to calculate the interaction between the proton and the electron. Not that difficult (by means of the Schrodinger equation), but when you try to do it in
connection with QED the story becomes somewhat more difficult.
So you can imagine how the situation is for higher mass atoms!
About the visibility of atoms.
It will never ever be possible to see how an atom looks like (in whatever way)! Maybe a computer-generated image, which is something different. Don't get fooled by people (writing for sensational magazines, to get the money coming in) who say they can. Look at this one:
First Atom made Visible
Again, don't let these popularizations fool you! You don't see the atoms, but an image of them, which is something completely different. Imagine you had the size in the order of atoms. How would you be able to see them? By shining photons on them? No. I think you are intelligent enough to see why this is impossible.
One more but the last thing. It is said in one answer here that QED is used in all calculations concerning the atom. NOT true.  @my2cts addresses this issue perfectly.
A: There are no final models in science, there's always room for improvement. And major paradigm shifts cannot be totally ruled out. However, we can be quite confident in our current model of the electronic structure of the atom, which is based on quantum electrodynamics (QED), which has been validated to very high precision.
Wikipedia has numerous orbital diagrams, including many animated ones. But you also need to read the text to understand how the diagrams work, and even then, it's not easy to understand what's going on unless you've studied some quantum mechanics, and are familiar with the basic concepts, and some of the mathematics. 
I'm quite fond of the animated diagrams in the section titled Qualitative understanding of shapes:

The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of standing waves on a circular drum
[...]  
The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the Heisenberg uncertainty principle for details of the mechanism).

We can make images & even movies of actual orbitals, but they're rather crude; the diagrams are better. I suppose that the images & movies are beneficially in that they demonstrate to the lay audience that the diagrams are valid, and not just some mathematical fabrication. ;)

It is not easy to appreciate exactly how electrons move inside the atom. Things at the quantum scale simply do not behave in the way we are accustomed to at the macroscopic scale, so our normal intuitions aren't much help when it comes to electrons. That does not mean that these things are incomprehensible, but it does mean that we can mislead ourselves if we try to apply classical notions to these decidedly non-classical entities.
So while electrons in atoms certainly have kinetic energy and momentum (including orbital angular momentum, apart from electrons in s orbitals), it's a mistake to ascribe any kind of classical trajectory to them. 
A: The most important thing is to understand the difference between orbit and orbital. Originally the atom was imagined with classical orbits (like planets around stars), but eventually QM was developed and now we talk about electron orbitals.

In atomic theory and quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom.1 This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as defined by the particular mathematical form of the orbital.[2]

Now today, we have basically three types of orbitals:


*

*Hydrogen like

*Slater Type

*Gaussian type
So basically the current atomic orbital models describe the electrons' existence around the nucleus as a probability distribution. Now you are asking whether these have certain shapes? Yes they do.

https://en.wikipedia.org/wiki/Atomic_orbital
