What exactly is Landau Damping? I would like to understand what exactly happens when Landau damping takes place. I have seen the mathematics of it. But I am not sure If I truly understand what happens when we are talking about Landau damping.
 A: A way of intuitive thinking about Landau daming is from the point of view of the electrostatic wave (or Langmuir oscillation). This will also clarify a bit why the whole mathematical apparatus is needed.
Short Intro to Electrostatic Waves
Without a magnetic field, in an electrostatic field you can consider the Vlasov equation and derive a dispersion relation for a wave,
$$
1 + \frac{\omega_e^2}{k^2} \int \frac{\partial v_x g(v_x)}{\omega/k-v_x} dv_x = 0,
$$
where $\omega_e$ is the electron plasma frequency, $v_x$ is the x-velocity coordinate and $g(v_x)$ is a 1d distribution that relates to the background distributions of all species (where $v_y, v_z$ is integrated out).
You can see that this is quite complicated and contains a pole at $k v_x = \omega$. Anyway, using a bit of hand-wavy assumptions (the pole is away from a region where the distributions are non-zero) you can integrate this and (after some manipulation) arive at the Langmuir wave dispersion relation (only taking electrons into account into $g$),
$$
\omega^2 = \omega_e^2 + 3 k^2 v_e^2,
$$
where $v_e$ is the mean velocity of the electron species (1st moment). This is basically a simple oscillation of the electrons with a frequency slightly higher than the electron plasma frequency.
Landau Damping
So now so good, the part where Landau damping comes into play is where you ask yourself if the integral should need a proper treatment. Then you have to make the integration in the complex plane and so on. You will get a modification to the above dispersion relation, but the main point is, that it becomes complex with the imaginary part or damping,
$$
\gamma \propto \partial v_x g(v_x) \Bigg|_{v_x = \omega / k}.
$$
This means, that there is damping which depends on the slope if the distribution $g(v_x)$. Here come the metaphors with e.g. the surfers into play. A surfer slower than a wave is sped up until he reaches the bottom of the wave, if he is faster the other way round. The surfer shoul be associated with a particle, it is loosing or gaining energy from the wave, depending on its velocity compared to the wave.
One important remark here is, that we used the Vlasov equation only, i.e. no collisions. Thus, Landau damping is classified as collisionless damping.
References
I mostly recommend Nicholson, Introduction to Plasma Theory (although it is quite old and possibly impossible to get nowadays). You may also find more in Chen, Introduction to Plasma Physics and Controlled Fusion (including the surfer).
A: This wikipedia article explains the physical picture rather well. Landau damping results from the energy exchange between the electromagnetic wave and the plasma particles, its main feature being the dependence on the distribution of the particles.
As an analogy, perhaps a bit far-fetched, one could compare Landau damping with the interaction of an electromagnetic wave with an active environment of a laser. As long as the population of the ground state is higher than that of the excited state, we always have more photons absorbed than emitted. However, if the populations of the ground and the excited states are inverted, the opposite happens and we achieve the amplification of the signal.
Such thinking about damping and amplification is rather comonplace - Landau's main achievement is describing (and predicting) it in the context of plasma. Note that this was quite commongly and literally used in various kinds of bulb devices throughout the XXth century, but less familiar nowadays (which is why the laser analogy might be more understandable.)
A: There are a few critical issues that I did not find in the other answers which are worth noting here.
Landau Interactions
Landau interactions are specifically tied to the term $\mathbf{k} \cdot \mathbf{v}$, where $\mathbf{k}$ is the wave vector and $\mathbf{v}$ is the particle velocity.  The physical interpretation of this term is that longitudinal electric field fluctuations exchange energy/momentum with particles along the direction of the wave vector.
I made an animated GIF to illustrate how Landau resonance can work for an electromagnetic wave propagating obliquely to a quasi-static, background magnetic field (red lines in below GIF).  The phase fronts are illustrated in blue and particles as little black dots.  Owing to the Lorentz force, charged particles prefer to follow rather than cross magnetic fields so for something like an electron to stay in phase with this cartoon wave, it will have to move very quickly along the magnetic field.

Note that $\mathbf{k}$ is the vector that is actually projected, not the vector associated with the phase speed.  It's a common misunderstanding that there is a phase velocity but it's kind of misleading, since it's really a phase speed along the $\mathbf{k}$.  This is relevant for velocity transformations as the phase velocity does not transform under Galilean transformation like a regular velocity.
The point of that tangent was to illustrate that the resonant speed required by particles in this cartoonish scenario can be extremely high parallel to the magnetic field compared to perpendicular, even when the phase speed itself is rather small.  This results because the magnitude of the projection of $\mathbf{k}$ along the magnetic field is tiny, thus the angular frequency over this projected magnitude is very large.
Cyclotron Interactions
Since electromagnetic fluctuations can exhibit both longitudinal and transverse oscillations, one can actually get two types of interactions.  Landau interactions are one, the other is called cyclotron.  Cyclotron interactions require that an electric field oscillate in phase (or close to) with the cyclotron motion of the charged particle.  Unlike Landau, cyclotron interactions can result from both longitudinal and transverse fluctuations.
To see the longitudinal process for cyclotron interactions, think of a linearly polarized electric field oscillating orthogonal to a quasi-static, background magnetic field.  If that field oscillates near the cyclotron frequency of a charged particle, it will accelerate the particle.  I have added an animated GIF I created years ago to illustrate this concept.  The top panel shows particle position in red and electric field in green while the bottom panel shows particle velocity in blue and again electric field in green.

Lastly, note that both cyclotron and Landau interactions work both ways.  That is, the fields can give energy to the particles (damping) or the particles can give energy to the fields (growth).
