What does it mean by a wave oscillating? It is in my textbook that a wave oscillates.If we see diagram of a wave , it shows a wave’s amplitude going up in + direction and then down , in - direction.

Amplitude is the maximum displacement of a wave from its mean position. I don’t that when we draw a wave or just in real life saying how it would look like , it will come in oscillating way. The diagram that we have drawn is not the wave structure but it’s amplitude variation.
When we say maximum displacement of a wave from its mean position , What does that mean exactly ? Like the wave comes ahead and then goes back , by looking at the diagram it goes to negative value.I am not able to understand it.Please help me in understand it.
 A: I'm not entirely sure where your confusion stems from but here's an animation of a travelling wave:

The length of the red arrow shows the amplitude. The amplitude is defined this way because when you work with the maths behind it it is the most convenient quantity to work with.
You can imagine this being a water wave. The $y=0$ axis is the height of the water when it's at rest. If you generate a wave by dropping a stone in the water or something the waves will look roughly like this. The blue line shows the height of the water so when the wave reaches $y=1$ it means the water is 1 unit higher than it would be when the water's at rest.
Here's an example of a standing wave:

This occurs when two waves of the same amplitude meet.
A: A wave is a propagating disturbance. "Of what?" would be anyone's next question and the answer is that it depends on what type of wave one is talking about. Let's consider a few examples.
I suggest you do the following experiment at home. Take a rope (or a wire or a string) and tie its one end to a door knob or a nail in the wall. Hold the other end in your hand and hold the rope tight. This is the rope's equilibrium position. See figure below.

Consider the specific infinitesimal section of the rope that you hold in your hand. What are the forces acting on this section? Well, there is the weight of that section pulling itself down. Then the rest of the rope is pulling it along the direction of the rope itself balancing your pull. The three forces cancel out and this section remains at rest. The section infinitesimally to the right of this section experiences three forces. The rope pulls to the right and slightly up. The section to the left pulls it to the left and slightly up and the weight pulls it vertically down and so on.
Now give a jolt to the section you hold in your hand and stop abruptly letting the rope slightly loose but not completely loose. What happens?

You disturbed the equilibrium. At the moment when you first apply the jolt, the section you hold gets an upward acceleration. The moment you stop applying that extra force this section decelerates and then eventually falls back. The section immediately to the right of this section is now disturbed off its equilibrium though. As the figure below shows this "disturbance" now travels along the direction of the rope. The disturbance itself is vertical and hence perpendicular to the direction of the wave which itself travels along the direction in which the rope is oriented. Therefore the fancy term for such a propagation is a transverse wave.

The rope sections themselves are not moving horizontally. They are in fact oscillating up and down from the mean original equilibrium position. The hump physically does not exist. It is a visual trick. If you focus on only one section of the rope you see it go up, then come down, and then remain still until the "hump" comes back and hits it again. The entire play here is of the various forces (weight and the tug from left and right) acting upon each section. Once a disturbance is introduced the wave remains until it dies out from dampening introduced by other physical factors. The height of the hump is the amplitude of this wave.
If you keep oscillating your end of the rope for a bit longer you will setup a more interesting wave with crests and troughs. The amplitude will be either the depth of the trough or the height of a crest. In an ideal scenario they may be the same. But in real world they may not and one can then take an average of the two or better still define amplitude as a function of space and time depending upon how much detail they want.
If you now take a slinky and tie its one end to the wall or doorknob and hold it taut like you held the rope before you can create a longitudinal wave by stretching one end horizontally and then letting it slightly loose so that it can go back. It is called so because the disturbance (i.e., the oscillation of each slinky section) is along the direction of the wave. You will observe a distinct "compressed" section of the spring travel across the length of the slinky.
Once again, you can break all the physics down to play of forces. The sections to the left and right of any chosen section of the slinky are pulling the slinky left and right and the disturbance in this case is displacement of an infinitesimal section of the slinky from its original rest position. The amplitude in this case is the difference in the lengths of a section when it is compressed and when it is not.
A: There are two (main) types of mechanical waves: transverse and longitudinal.
Longitudinal waves, for example sound waves, are also known as compression waves because the direction of propagation of the wave is parallel to the direction of oscillation of the constituents of the medium through which the wave is passing.
Transverse waves, for examples waves on a taut string, are waves in which the direction of wave propagation is perpendicular to the direction of the displacement of a particle being moved by that wave.
So it seems you're asking specifically about transverse waves.

When we say maximum displacement of a wave from its mean position , What does that mean exactly ?

The simplest kind of transverse wave is a sinusoidal plane wave, which is modeled as a sinusoid that has frequency and amplitude. The "maximum displacement" from the mean position is the amplitude of the wave. This means that the wave moves a particle from the mean position to the maximum position in an amount of time that is the reciprocal frequency (i.e. the period).
Here are some nice visualizations of different kinds of waves.
A: "Oscillation" literally denotes something moving in one direction, then moving back. I think you'd agree that a physical pendulum oscillates--it moves right, then left, then right.
Physics uses the term by analogy to mean a quantity that "moves" back and forth (as you can visualize using Cartesian coordinates). So, in an electromagnetic wave, the magnetic field strength increases, drops to zero, increases (the other way), decreases, etc. In a sound wave, the pressure increases, decreases, increases ...
I hope that is helpful.
A: I'm having trouble discerning what the source of your confusion is, but I have a feeling it has to do with the apparent motion of the wave front. So I don't know if this will help, but, in a transverse wave, what's oscillating are the individual particles that comprise the wave-carrying medium - they just go up and down, even though the wave itself is traveling:


Both of these diagrams show the wave as it looks in space (the horisontal axis is the extent of the wave; they show the spatial "structure of the wave", as you've put it); the time is represented by the animation.
You can see that each particle is oscillating around a mean value (a place where it would rest if there was no wave), just like a linear harmonic oscillator. The displacement is just how far it is from that mean value line.
A: This is very helpful for me as I do struggle in physics on this subject as I used to get confused with all the different names.
