Why do we think of light as a wave? I've read that light travels in a straight line and has a wavelength of 400nm to 700nm. But I don't understand why does it have a wavelength and what creates its wavelength? I agree with the concept of sound which also has wavelength, thus called sound waves which are created by the vibrational movement in air. I'm not aware of calling light a wave. Does the light vibrate too? If so, then how?
 A: I want to give you an example of how light can created. This has been helpful for my students when it comes to visualizing light as a wave. This is not the only way that light can be created, but it's a way that is a little easier to visualize and may help you understand why it can be seen as a wave.
Imagine an electric charge. This charge creates an electric field around itself, which is what allows it to put a force on other charges.
If we move that charge, the electric field associated with it will also move. However, and this is really key: the electric field does not adjust instantaneously. It takes time for the field to "catch up" with the new position of the charge. There's a great simulation of that at this link. It's a little exaggerated, but it'll show you the basic idea. You can see that wiggling the charge creates a disturbance in the electric field that looks very wave-like. 
There is also a magnetic part to this, which is more complex and is not shown on that animation.
Moving charges are how we create radio waves, which are a form of light. (They're not the visible light that you were referring to, but there are a lot of different kinds of light.) Radio waves are made by moving electrical currents up and down the broadcast tower via a circuit that's designed to do so. 
It took a while for people to realize that radio waves, visible light, x-rays, and many other things are all really different forms of light. Once we realized that, it was clear that we could represent visible light with a wave - an electromagnetic wave, with a wavelength, frequency, amplitude, speed, and all the things that waves normally have. The wave is not "light vibrating," the light itself is vibrations in the electric (and magnetic) field.
I should also mention that @Robert is correct. Light can be modeled as a wave because it behaves like one under certain circumstances. It can also be modeled as a particle (the photon) or as a collection of waves called a "wave packet." 
I hope that helps!
A: Light is a electromagnetic wave. We talk about the dual-nature of light since in some instances it displays properties of a wave and other times as particles, each in a different state of quanta. 
When a metal is heated it will emit photons as the electrons shift their energy levels. Since electrons are emitted from a metal at a certain frequency of light it indicates that light consists of a particle called a photon. Increasing the intensity of light increases the rate of electron emission. This is not a property of a wave. 
However when light hits a transparent object at a angle it will be refracted (except at 90 degrees). This is a property of a wave. 
Therefore light is seen as electric waves and magnetic waves oscillating at right angles from each other. Light is always either a wave or a particle but never both at the same time. Welcome to the dual-nature of light.
A: Light is a wave - an electromagnetic wave.  Radio waves and microwaves are also  electromagnetic waves, they just have different wave lengths.  Wikipedia has a nice picture showing the electromagnetic spectrum

why does it have a wavelength...

It has a wavelength because there is physical space between the peaks of the waves - it is a real, physical, wave. Just like water waves and sound waves, you can do "wave things" to light waves, such as send them through diffraction gratings and see the interference.

and what creates its wavelength

Whatever creates the light gives it energy, and the wavelength is proportional to that amount of energy:
$$\lambda = \frac{hc}{E}$$

Does the light vibrate too? If so, then how?

Sound waves are energy waves that compress matter - you can't have a sound wave in a vacuum.  Light waves are energy waves too, but they don't need matter to go forward.  That's why we can see sunlight, but we can't hear the sun.  (And that's why in space, no one can hear you scream.)
This subject can get really complicated really fast.  Because although light is a wave, it is also a particle.  A lot of really smart people have been scratching their really smart heads over that, and will be for a long time.
A: To begin with you just bother about classical limit of physics (forget quantum mechanics). When you solve the Maxwell's equations in presence of no charge, you will end up with an equation which is similar to a wave equation. If you see history of partial differential equations, mathematicians have found a bunch of equations like "Heat Equation", "Wave Equation", "Poisson Equation" etc. And the maxwell's equation in no charge and electric current matches exactly with the wave equation. This led to believe that light is made up of a wave of magnetic field $\vec{B}$ oscillating mutually perpendicular to wave of electric field $\vec{E}$.
A: Why is very difficult to answer...
The speed of a wave is called celerity because the media in which the wave take form does not move but nature found a way to propagate, dissipate energy without moving the whole matter : the wave.
For light there is for example an electrical perturbation (let's say a spark) in space then Maxwell equation tell us how the electrical pertubation generate magnetic perturbation which generate magnetic perturbation and so on. The pertubation is then of electromagnetic nature and the speed of propagation is c.
I hope I answered your question, if you need me to go further let me know because question begining by the word why are the most interresting but the most difficult to satisfy.
A: Currently light is thought of both as a wave and being made up of particles (photons), because as Robert mentioned in his answer, certain phenomena require modelling light as a wave to explain (interference, diffraction etc.), and others require photons (such as the photo-electric effect).
Why do we think of light as a wave? Because modelling it as a wave correctly predicts a large range of physical phenomena. What is the 'wave'? Maxwell's Equations tell us that the wave is perturbations in the electromagnetic field. It is this field that is 'vibrating', or oscillating.
A: In classical physics, light (visible and invisible) is mathematically modeled as an electromagnetic wave, i.e., waves in the electric and magnetic fields.  
Electromagnetic waves are not limited to visible light.  For example, radio waves are simply very long wavelength "light".  X-rays and gamma rays are very short wavelength "light".
Light doesn't vibrate.  Rather, the electromagnetic field supports propagating "disturbances".  For example, if an electron were to suddenly accelerate, the disturbance in the electric and magnetic field associated with the electron propagates outward from the location of the electron with a speed of c, the speed of light.  This is somewhat analogous to a disturbance in the air due to, e.g., a loudspeaker that propagates outward at the speed of sound.
Of course, imagining "vibrations" in the electromagnetic field is not easy like imagining "vibrations" in air.  For years, physicists thought there had to be a substance, called aether, that filled all of space and was the medium for the propagation of light waves.  It was the failure to detect the aether that led to the development of the Special Theory of Relativity. 
A: Light is not a wave.  If you (for instance) shoot particles of light called photons from a laser (photon-gun) at a wall with two holes in it and for some odd reason you decide to fire one photon every 1000 years you will find out that at the other end the detector (some fancy sort of polaroid) has an alternating series of vertical dark stripes and vertical light stripes.  Thus you have what does look like an interference pattern on the screen.  That is why it's more correct to say that light is a particle (the photon) which can "behave" like a wave. However as previous commenters explained if you don't want a deep insight into the nature of light you can also think of it in a classical (non quantum mechanical) way as a vibration of the electromagnetic field but in my opinion that response is not only less deep but also a little more abstract! The abstractness of quantum mechanics comes into play in the "why" of why there's an interference pattern but if you understand quantum mechanics already you don't need to also understand all of electromagnetism to understand why light behaves like a wave because EVERYTHING has a wavelength associated to it.  Electrons also behave the way I described when you fire an electron-gun at a wall with two holes. 
A: If visible light is part of LIGHT, which has a wide spectrum, including gamma, x-rays, sound, then why is it said that Light travels at a constant speed, when clearly it means visible light, since sound does not travel at the same speed as visible light, correct?
