What exactly does a sinusoidal wave represent in relation to sound waves? So I'm very new to wave mechanics.
Here is a picture of a sound wave:

Now what I'm not able to comprehend is what exactly this wave represents. Does it represent the oscillation of a single particle in a medium? Or does it represent the wave traveling through the medium as a whole? If this is the case then what exactly does the up and down displacement of the wave on either side of the mean position denote? Is it all the particles oscillating?
It would be of great help if you could break this down to me, and actually tell me what all aspects of sound propagation, this wave represents.
What I want to know exactly is this: when the y axis represents wavelength, does it mean that, a particle of the medium oscillates, then the subsequent particle oscillates and so on, and each oscillation of the subsequent particles is represented by one Crest and trough together?
 A: In short, pressure.
For an acoustic wave (a wave traveling through air), this figure will most commonly refer to the pressure.  But that's mostly because pressure is the thing that's easiest to measure in acoustic waves.  The velocity of the particles also follows the same graph so it could be a graph of velocity of the particles as well.  (That is, the vertical axis of the graph could represent either pressure or velocity; although, as an aside, if you graph both together the graphs will be shifted along the axis).
For acoustic waves, you can think of the particle oscillations occurring in the direction of the wave's travel and driving a bunching of the particles and creating the pressure and, vica versa, as the pressure differences driving the velocity of the particles.  These coupled disturbances are self-supporting and create the sustained wave propagation.  (But note that the individual particles are not moving a full wavelength, instead, they all move a small amount but in a way that creates bunching and rarefaction of the particles.  Here's a video that shows this, where the main point is that if you follow a single particle, it doesn't move with the wave or move a full wavelength but only oscillates around a local position.)
Furthermore, the horizontal axis could be either space or time, since for a traveling wave, you will see a similar oscillation if you stay at one place and watch the wave go by, or take a snapshot in time and look at the variation in space.  The figure shows $\lambda$ which is the common symbol for a wavelength, so it's space in the figure, but it could just as well be time, in which case it would show $T$ which is the usual symbol for the period of oscillation.
More generally, these traveling sinusoidal oscillations occur very commonly.  The primary reason for this is that they are the mathematical solution to the equation of a wave traveling through linear media, and at least for small amplitude disturbances, most media appear linear.  So, you will find that most traveling waves are described this way, regardless of the internal mechanisms of the wave.  When you're reading the description of the acoustic waves, you can understand it in terms of pressure, but keep in mind that it will apply much more generally, and that's probably why it's being presented in a fairly abstract way (which is also why I keep saying "acoustic waves", because most of the detailed mechanics aren't generally true, but the sinusoidal nature remains).
A: 
Now what I'm not able to comprehend is what exactly this wave represents. Does it represent the oscillation of a single particle in a medium? Or does it represent the wave traveling through the medium as a whole?

The sinusoidal curve itself can represent both; however, in this particular example, it represents the shape of the wave throughout the medium as a whole (so, throughout space) at a particular instant in time. You can tell that this is the case because the image has the annotation for the wavelength ($\lambda$) along the horizontal axis (the wavelength is the spatial length between two crests or troughs).
The image you posted shows one "frozen" moment in time; the vertical axis is the displacement of the particle at the corresponding point in space.
Now, if you "unfreeze" time, you get something like this:

Note that each individual particle stays "in its lane" and oscillates around a midpoint (you can track one of the red ones). As you can see, the displacement of each particle changes in time. In fact, each particle behaves like a linear harmonic oscillator, so if you pick one particle, and plot a graph of its position against time, you also get a sine curve. The animation below shows the displacement of a single particle at any given point in time:

Actually, you can plot both at once. In the image below, imagine that the vertical plane on the left (the "space" plane) is sliding along the time axis (because of the passage of time); its intersection with this wavy surface produces the traveling wave:

Now, here's the kicker; all kinds of waves can be described this way. The vertical axis just abstractly encodes the displacement from the equilibrium position - it doesn't tell you anything about the orientation or nature of that displacement. So, there's no real reason why the displacement has to be up-and-down. Sound waves are longitudinal waves, which means that the displacement happens in the direction of travel (the particles oscillate left-and-right). See this beautifully animated illustration that i found here. The black arrow in the center shows the displacement from the equilibrium position (marked by the vertical dashed line); the graph at the top plots this distance on the y-axis (positive values mean that the arrow is pointing to the right, negative that it is pointing to the left).

Note also that sound can be described as a pressure wave as well - the curve at the bottom - where "displacement" corresponds to local deviation from equilibrium density (compressions and rarefactions).
A: Typically, when we look at a travelling sine wave, as your diagram shows, we think of particles moving up and down, this is what is called transversal motion. It is transversal because the oscillation of the particles is perpendicular, that is transversal, to the direction of travel.
However, sound waves are not transversal waves, they are longitudinal waves of compression and rarefaction. Here the particle oscillates in the direction of travel.
A: What you have shown in the picture is not a wave but rather a state of deformation of a sound carrying medium (i.e. a description of what you have referred to as the "whole medium") at a specific instant in time. Just like for Newtonian point mass dynamics you also need to specify initial velocities (deformation velocities) in addition to initial positions in order to describe what happens next.
In a carrier medium (e.g. steel or air) there can be many waves that travel in many directions all at the same time, often without disturbing each other. This is called the superposition principle (which is only true if the medium is said to react linearly). But very special waves that only travel in one direction are called plane waves. This means that the propagation direction defines the set of planes that are perpendicular to it (of which there are infinitely many, of course) and if you look only at one of these plane, everything about the deformation state and its time derivative ("velocities") looks the same.
In addition to the direction of propagation, the wave also has "directions of polarization" (I don't actually know if this is the "official" general term for this in continuum mechanics). This means, it can have different kinds of deformation with respect to the direction of propagation.
For a fluid, the only type of deformation that is possible is compression/dilation. This is how air conducts sound. It means that for one instant in time, and for a plane wave, compressive deformation (and with it, pressure) varies along the direction of propagation in the way you have shown it in the picture. Perpendicular to this direction (the planes) pressure is constant. Since this means that particles effectively move back and forth in the direction of propagation, these waves are also called longitudinal waves.
For solids, that don't have any preferred direction (and which are thus called isotropic), there is also the possibility, that particles move perpendicular to the wave propagation, and hence these waves are called transversal waves. The corresponding deformations are called shear strain and the forces are called shear stress (which can be considered "a different kind of pressure"). You can imagine that shear are the forces that deform a rectangle to a parallelogram.
Just like there is a superposition principle for waves of different propagation direction, there is also superposition of different "polarizations", if the medium is linear and the different wave types don't influence each other.
For non-isotropic solids (for example perfect crystals), things get a little more complicated. The deformation state is then described by a set of 3 x 3 numbers, called the strain tensor (a "matrix"). Correspondingly there is a 3 x 3 stress tensor. Non-isotropic solids conduct sound differently in different directions. But that only as a side-note.
Alternatively to describing the current state of a medium by deformation, there are many other, more or less equivalent way. For example, instead of deformation plus change of deformation, you can also describe it by velocity and acceleration, or by velocity and pressure. In a linear medium, all these descriptions will principally show a corresponding wave behavior. So it doesn't matter much, what the described quantities are, if you just want to understand waves.
The point is now, that a wave is not only described by its spatial behavior (which you see in the picture), but also by its temporal behavior. One way of expressing a plane wave is by a sinusoidal function of time and the distance in the direction of propagation (let this be the x axis):
$$\psi(x,t) = A\cdot \sin(k\cdot x-\omega \cdot t + \phi_0)$$
where $\psi$ can be, for example the degree of compression. The fact that only $x$ appears in the sine, but not $y$ or $z$, mostly represents the fact that it is a plane wave travelling in the x-direction. Hence, everything looks the same in the $yz$ plane. If you take a snapshot at a fixed time $t=t_1$ (like in your picture) then you obtain a simple spatial sine function (the wave of the "whole medium")
$$\psi_1(x)=\psi(x,t_1)=A\cdot \sin(k\cdot x-\omega \cdot t_1 + \phi_0)=A\cdot \sin(k\cdot x + \phi_1)$$
and hence you see justified how the picture looks. The wavelength $\lambda$ is the distance between two parallel planes of this plane wave snapshot pattern, that look exactly the same, and it is determined by the period of this sine function. You can find it by remembering that the period of a "raw" sine is $2\pi$:
$$k\cdot x = 2\pi$$
and therefore
$$x=\frac{2\pi}{k}=\lambda$$
The quantity $k$ is also called the wave number (for some reasons that are rooted in optical spectroscopy, and how results are usually presented there).
On the other hand, if you consider what happens at a particular location in space $x=x_2$ then you obtain a temporal sine function (more or less what you have referred to as the oscillation of a single particle, which is not shown in the picture; but of course you will always be describing many particles by that equation, and they don't stay in place, but their collective movement is the oscillation)
$$\psi_2(t)=\psi(x_2,t)=A\cdot \sin(k\cdot x_2-\omega \cdot t + \phi_0)=A\cdot \sin(-\omega \cdot t + \phi_2)$$
The "repetition time" (commonly called period) $T$ of this temporal oscillation is again determined by the period of the "raw" sine, i.e.
$$\omega\cdot T=2\pi$$
and therefore
$$T=\frac{2\pi}{\omega}$$
More often one expresses this by the frequency
$$f:=\frac{1}{T}=\frac{\omega}{2\pi}$$
usually given in the unit Hertz (Hz).
A: It all depends on what the axes represent. If the vertical axis is the displacement of a particle and the horizontal axis is time, then it is showing the oscillation of that particle. If the vertical axis is air pressure and the horizontal axis is distance along the wave, then it is a snapshot of the air pressure along the wave at a given time.
Edit: By "air pressure" is meant the deviation of air pressure from what it was before the wave came.
A: Rather than guess what someone else's graph means, first plot your own graph. On the horizontal axis you can put whatever is suitable for the thing you are plotting on your graph, and similarly for the vertical axis.
For example, in a sound wave the particles in the medium (e.g. it could be air, but you can also think of a solid medium) are moving to and fro a little bit, in the direction along which the sound wave is moving. The particles don't move very far. You could compare it to a line of people in a long queue, where each person keeps their feet still while they swing forward and back and push gently on the next person. In consequence of this motion, the density at each spatial location oscillates, and so does the pressure. I think that density is easier to think about than pressure, so let's plot the density first. You can either consider one point in space and plot the density at that point as a function of time, or you can consider a line of points and plot the density along that line, as a function of distance, at some moment in time. Either graph will have the form of a sinusoidal oscillation.
After plotting density, you could plot pressure. The graphs will look the same. Then you could plot something that takes a little more thought: the displacement of a particle in the medium. Each particle is moving to and fro, so it has a displacement from its equilibrium position. You can pick one location and then plot the displacement of the particle whose equilibrium position is at the location you picked. You would be plotting it as a function of time. It is a sinusoidal oscillation again.
Finally, you can if you like plot the set of displacements of all the particles that lie along some chosen line, at some given moment in time. This would be a snapshot of the waveform. It will look like a sinusoidal curve again.
When you plot as a function of time, the peaks in the graph are separated by the period of the wave. When you plot as a function of distance, the peaks on the graph are separated by the wavelength. The graph shown in the question has no axis labels, which is bad practice, but since the symbol $\lambda$ is shown between the peaks I deduce that it is plotted as a function of distance along the wave. As for the $y$ axis on the graph, it could be density, or pressure, or displacement, or indeed possibly other things such as velocity, rate of change of pressure $\ldots$ the list is endless.
