How exactly is white light a combination of several wavelengths? I have read that light is an electromagnetic wave. Every ray of light has a specific wavelength. The colour perceived by any observer is dependent upon the wavelength of the incident light.
What I don't understand is that how do electromagnetic waves of different wavelengths combine to form a single wave of another wavelength? Simply put, I have the following two related questions to ask: 


*

*When we look upon a, say, completely white object, what is the composition of the individual rays that strike our eyes? Are those rays waves formed by the addition of waves corresponding to individual wavelengths that constitute white colour? I get that white light is composed of all wavelengths of visible light, but how are those wavelengths combined into a single unit which we call white light?

*If it is so, then how are prisms able to disperse light into its constituent colours?


Also, as a side question, how does all of this relate to light being composed of photons?
 A: The OP's restated question is 

I get that white light is composed of all wavelengths of visible light, but how are those wavelengths combined into a single unit which we call white light? Are they just sticking to each other like a bundle of sticks, perhaps with spacing in between them?

It appears that the OP is visualizing a light "wave" as a "ray" comprising a line with tiny wiggles -- maybe something like a guitar string.
In classical electromagnetic theory, actually a light wave is more like a wave on water (though not confined to a 2D surface). A water wave can be a single frequency, in which case the spacing between ripples is constant and the shape of the ripples is a perfect sinusoidal wave.  If the water wave is comprised of multiple frequencies, the spacing and shapes of the ripples vary from place to place (and of course they vary in time as well).  
If we were to draw lines in the direction the ripples are moving at each point, the lines would correspond to what we think of as "rays"; but those lines are not objects, they are only a way to describe the motion of a point on a wave.
Light waves (from the classical perspective) similarly correspond to ripples in electric and magnetic field amplitude and direction, within a 3D volume. "Light rays" are not real objects, they are just a way to describe the motion of the wavefronts of light.  A mix of colors/wavelengths/frequencies amounts to a mix of spacing between the ripples, and to variations in the shapes of the ripples. This answers the first part of your question.
The second part of your question:

If it is so, then how are prisms able to disperse light into its constituent colours?

This is very different from the first part of your question. So: remember that a "ray" is really just a way to describe the motion of a point on a wave.  This works fine for waves that never change shape or spacing, but gets ambiguous when the wave is not a perfect sinusoid.  Suppose each "ripple" has an extra bump on it.  In that case we can imagine that the overall (3D) wave is actually the sum of two (3D) waves, where one wave has half the wavelength of the other. 
Every light wave can be resolved into a superposition of different-wavelength waves, which may be traveling in the same direction or in different directions.
When light of a single wavelength enters a prism, its direction changes because its speed changes.  But the speed of light in glass is dependent on the wavelength, so the new direction is different for each wavelength.  In the case above, when a wave is composed of two waves having different wavelengths but the same speed, incident on the prism from the same direction, the longer-wavelength component ends up moving in a different direction from the shorter-wavelength component.
It's still all one complicated wave, but the wave segregates into different regions where the ripples have different spacing.  "Rays" drawn to describe the motion of the ripples will indicate that the shorter-spacing ripples are moving in a direction different from the direction of the longer-spacing ripples.
A: David White's comment is correct, and I think the existing answers are confusing the point. The poster asks:

How are those wavelengths combined into a single unit which we call
  white light?

They aren't. There is no "unit" called white light. Our eyes have receptors for light of three wavelength ranges (graph here), and a collection of incident photons that excites each of them roughly equally is what we sense as "white." It's similar to blue and yellow light giving the sensation of green. There is no "wave interference" that makes a blue wavelength and yellow wavelength combine to green; it's solely an aspect of human perception.
While it is physically possible to make "white" light with electromagnetic wave packets that are very short, and so spread out in frequency throughout the visible range, this is neither necessary nor typical.
A: To understand this you need to learn about the fourier transform. It describes how a function of time (in your case the time dependent electric and magnetic field strength) can be decomposed into its constituent frequencies. 
To me your question seems similar to asking 
"How do vertical and horizontal velocity of an object combine to form a single velocity in a different direction". 
They combine by just adding the vectors up. Of course the result points in the same direction. It's just a different description of the same thing.
Similarly

how do electromagnetic waves of different wavelengths combine to form
  a single wave of another wavelength?

When you have two electric fields you can add their vectors to get the resultant field. So they combine by just adding the vectors up, thats it. Of course they don't "form a different wavelength". The result has the same wavelengths.
A: I am going to talk about visible light, I assume you are asking about that.
EM radiation between 380 and 760nm is detected by the human eye and preceived as visible light.
White light is a combination of different wavelength photons in the visible spectrum.
The reason we perceive it as white is because our eyes contain sensitive cones and rods, that are sensitive to different wavelengths. 
When white light (that is made up by all different wavelength photons in the visible spectrum) hits our eyes, all the cones and rods get activated (these are sensitive to different wavelengths, red, green, blue), and in our perception, they are combining them to a single light that is perceived as white.


The perception of "white" is formed by the entire spectrum of visible light, or by mixing colors of just a few wavelengths in animals with few types of color receptors. In humans, white light can be perceived by combining wavelengths such as red, green, and blue, or just a pair of complementary colors such as blue and yellow.[4]


https://en.wikipedia.org/wiki/Color_vision
A good example is Sunlight, that is white (contrary to popular belief it is not yellow), that is built up by all the different wavelength photons in the visible spectrum.
Light changes speed as it moves from one medium to another (like from air in your case to glass of the prism), this speed change causes the different wavelength photons to enter the new medium at different angles (Huygens principle). This is how a prism is able to separate the different wavelength photons inside white light.


The degree of bending of the light's path depends on the angle that the incident beam of light makes with the surface, and on the ratio between the refractive indices of the two media (Snell's law). The refractive index of many materials (such as glass) varies with the wavelength or color of the light used, a phenomenon known as dispersion. This causes light of different colors to be refracted differently and to leave the prism at different angles, creating an effect similar to a rainbow. This can be used to separate a beam of white light into its constituent spectrum of colors.


Photons are elementary particles of light, part of the SM. It is the quantum of the Em field and the force carrier of the Em force. The QM photon and its herd builds up perfectly the classical Em wave and the two theories work together perfectly.


In this link there exists a mathematical explanation of how an ensemble of photons of frequency ν and energy E=hν end up building coherently the classical electromagnetic wave of frequency ν.
It is not simple to follow if one does not have the mathematical background. Conceptually watching the build up of interference fringes from single photons in a two slit experiment might give you an intuition of how even though light is composed of individual elementary particles, photons, the classical wave pattern emerges when the ensemble becomes large.


Please see here:
https://physics.stackexchange.com/a/90649/132371
