What exactly are light waves? We know a sound wave is a disturbance that moves through a medium when particles of the medium set neighboring particles in motion. And using the pressure variations we can plot a 
pressure/time graph which looks like a 'wave'in the drawing sense.
What do we mean when we say light is a wave and particles? using the above system, can we  say that light is the disturbance that moves through a medium of photons( forgive me that makes no sense). When we draw a light 'wave', what does it mean? a photon goes up and down?
Please keep the answer as 'not complicated math' as possible.
 A: It may be useful to start this explanation from the origin of a light wave: an oscillating charge. Start with the idea that a stationary charge is surrounded by an electric field, then imagine wiggling that charge up and down. Now the field lines will turn to wiggles instead of straight lines. Those wiggling field lines are the electromagnetic waves we call "light" (see animation). It turns out (as said in another answer) that the electric and magnetic parts of the wave help each other and let the wave propagate in free space very efficiently. Now, imagine what happens when that wave hits a stationary charge (call it the "detector" charge): the waving electric field will pull the charge up and down and the "detector" charge will start to oscillate.
If these charges are the electrons surrounding atoms, then you start to see where light comes from and how it gets detected. We also see how the wave travels through vacuum: there is no medium that carries it. This is one of the remarkable features of electromagnetic radiation. 
As for photons, they are a tricky concept that still sparks debates between experts.The photon is a representation of light that is only useful when the light beam is very weak (so a photo detector makes distinct clicks) or is specifically prepared in a special state with exactly one (or exactly $N$) photons.
A: The light we see with our eyes is electromagnetic radiation, very well modeled by Maxwell's equations.


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together.

As the other answer states that was fine until the photoelectric effect showed that light is composed from zillions of photons, elementary particles in the underlying quantum mechanical framework from which classical behaviors emerge. In the link this is demonstrated but it is not simple mathematics, it needs graduate courses in physics and math, probably in your future studies.

When we draw a light 'wave', what does it mean? a photon goes up and down?

A zillion photons join up in phase to create the up and down variation of the classical electric and magnetic fields.
The way the classical wave behavior emerges from the zillions of photons is not simple, it has to do with the quantization of Maxwell's equations. I hope this helps in your future plans for study. 
A: Since this was not stated yet, I would just like to give my stance on it.
All fundamental particles can be seen as excitations of fields. This is true for photons, electrons, neutrinos, etc. Do these fields need a medium in which they propagate? Not as far as we can tell. Everything we see and experience are excitations of these fields, a single one of which we call a particle. This is a photon for photon (electromagnetic) field, an electron for the electron field, etc.
These fields have different properties such as their mass and they can interact with each other. If a particle is massless (rest mass if this is not implicit), it has to travel at the speed of light.
The spreading of light is definitely not the waves spreading in the background of photons, but the spreading of excitations in the field. No medium is required for the field to live in if we regard the fields as the fundamental quantities.
For some (massive) fields, the quantized nature is much more apparent from our everyday experience to us than others. But the quantization is a fundamental property applicable to all of them.
How do we reconcile the wave and particle picture? I would claim that the particle picture is a construct of our minds coming from our everyday experience of the world. 
Don't get me wrong: quantization is a fundamental property and if you want to call this 'particles', I guess that is a matter of nomenclature.
But we should just think in terms of excitations of quantized fields. Some excitations, e.g. some Fock state, have more 'particle-like' properties, other such as Glauber coherent states for bosonic fields more 'classical wave'-like properties.
A: Einstein once compared the photon with a famous person (sorry I forgot the name) who changed confession at young age and returned to its initial confession before he died:
Light behaves as a photon  at the starting point and at the end point, and it behaves like a wave during its travel.
By the way, the light wave is not going up and down, it is not a "wave" in space", a light ray is straight, taking always the most economic path.
A: Light waves are exactly a theoretical explanation of light radiation. Propagation of waves of electromagnetic fields is a good theory that works for low frequencies, but as Einstein showed (and was Nobel prized for) the photoelectric effect can only be explained if electromagnetic radiation is emitted as directed quanta of energy.
I guess that experienced light, as an elementary phenomena, neither is of the form a geometric wave nor a geometric particle. 
One thing about light that I think is worth to notice is that if movement is expressed as 
$\displaystyle\gamma^{-1}=\sqrt{1-\frac{v}{c}}$, then the motion of light become a multiplicative zero, which give an algebraic "explanation" of the composition of velocities:
Suppose $\displaystyle m_k=\sqrt{1-\frac{v_k}{c}}$, then the composition of the "movements"
$\displaystyle m_1\star m_2=\frac{m_1m_2}{\left(1+\sqrt{(1-m_1^2)(1-m_2^2)}\right)}$
For light the "movement" is $m_c=0$ and that's why it's a limit: $0\star m=0$. :)
