Why don't electromagnetic waves need vacuum to move? I understand that electromagnetic waves are caused by the oscillation of electric and magnetic fields. But how? Are there magnetic and electric fields in air, or in vacuum? How does this oscillation form a wave? Sorry, I am a new at this site, please simply put you answer so that I can understand.  
 A: To answer the title question: it's simply an experimental fact that EM waves don't need a material medium (made of atoms jostling one another) to propagate in. They are simply different from waves that are elastic disturbances in collections of atoms. Countless experiments demonstrate the transfer of energy, momentum and angular momentum across vacuums by electromagnetic waves.
If this seems a little weird, it is possibly because you are confusing the notions of empty space and "nothingness" at some level, and you may gain some satisfaction from the knowledge that modern physics thinks of the vacuum as a material thing. Space is made of quantum fields, and "empty space" is simply a shorthand for a region wherein those constituent quantum fields are all in their ground state. In this sense, the vacuum, too, is very much a medium, although it is a Lorentz-invariant one, because here the wave equation transforms in a Lorentz-invariant way. This is in contrast with elastic waves in a material medium, where there is a "privileged" frame, i.e. that at rest with respect to the medium they move in, and the wave equation takes on a different form for observers in different motion states relative to the medium. 
To answer what it is that vibrates: just as it is something about the state of the material medium (local stress, strain and other measurable quantities) that vibrates in the propagation of an elastic wave, so too is it the state of the vacuum that vibrates: measurable influences on test charges betoken the oscillating state of the passing wave. Indeed, the presence of an electromagnetic wave shows that the EM quantum field is not in its ground state in the wave's region of influence, and so, strictly speaking, the space influenced by the EM wave is not a perfect vacuum, because it is in an excited state by dent of the wave's presence.
A: 
I understand that electromagnetic waves are caused by the oscillation of electric and magnetic fields. But how? Are there magnetic and electric fields in air, or in vacuum? How does this oscillation form a wave? Sorry, I am a new at this site, please simply put you answer so that I can understand.

This is a very old question, and it got harder to figure out when it was realised that between the Earth and the Moon for example, there was a vacuum, so we longer had any crutch/excuse to lean on, such as the atmosphere, to somehow  explain  the propagation of the reflected light of the  Sun from the moon to the Earth. Or directly  from the Sun to the Earth.
You could look at this two ways. You could say that Moon's reflected  light is composed of photons, particles of light, which to put it crudely, don't need a  medium to travel through any more than the Earth needs a medium to travel around the Sun. The Sun fires photons out like bullets, they bounce off of the Moon, and enter our eye.
Another, more subtle and more interesting way of looking at the process, is to take the Maxwell equations outlined in the other answer, which treats light as a wave made up of an electric field and a magnetic field, which vary in space and time, but in such a way that they create/maintain each other as the light wave propagates through vacuum.
Even Maxwell was convinced that some medium was needed to carry this light wave, saying in the Encyclopaedia Britannica around 1860 that, in effect we could be sure that such a medium (the aether) was available to carry the wave.
Nowadays, the aether is seen as a non existent solution, a (more than) slightly desperate attempt to explain something that was already explained in Maxwell's own equations.
You ask for a straightforward answer, so I won't include the equations, just one standard drawing of a electromagnetic wave.

Image Source: Hyperphyics
What you need to take into account is that this is a drawing illustrating, not the reality as you would see in a water wave when you throw a stone in a lake, but a drawing designed to illustrate abstract math. It is misleading if you think of it as a conventional wave,  but it is possibly more understandable as an analogy, rather than than asking you to follow the equations.
Three related  examples to think about. The charge from an electron can be felt without needing something to "carry" the charge, much the same way as two magnets will attract each other in a vacuum, also the force of the Earth's gravitational field holds the Moon in orbit, even through there is nothing to carry this force. 
I am going to guess that you might take these for granted, but in their own way they are variations on the question you are asking here.
The point of the drawing above is to try and illustrate that as the electric and magnetic fields comprising the wave travel through space, they keep the wave in shape, as a changing electric field induces a magnetic field and vice versa. (Which is stressed in Maxwells equations). If you look at the drawing, you might see how this works out for a light wave.
A: If you have an electric field ${\vec{E}}$, say, due to a charge density ${\rho}$ and the charged body is in motion then from Maxwell's equation $$\vec\nabla\times\vec{H}=\vec{J}+{\mu_0}{\partial\vec{D}\over{\partial t}}$$
i.e., the motion of the charges immediately produces a magnetic field $\vec{B}$ which, the magnetic field, in turn produces an electric field given by  $$\vec\nabla\times\vec{E}=-{\partial\vec{B}\over{\partial t}}$$ So you need nothing but the Maxwell's equations for your EM to propagate; by saying EM waves need a vacuum to propagate you're actually implying that EM waves need no medium to propagate for vacuum in classical physics means absence of any medium.
A: There is the classical physics framework, where Maxwell's equations describe the propagation of electric and magnetic disturbances, as described in the answer by Sayontön Vöttacharjo .
Present day physics has come to the conclusion that the underlying level of nature is quantum mechanical. From this framework large scale classical physics behaviors emerge, in a similar way that thermodynamic equations emerge from the underlying statistical mechanics ones.
The electromagnetic waves are built up by an enormous number  of photons in superposition. Photons are described by a complex wavefunction which obeys a quantized Maxwell's equation. In the complex expressions the electric and magnetic fields are oscillating. This is not observable when a single photon is detected, as a photon just has energy, mass zero and a spin either in +1 or -1 in the direction of its motion. The "waving" is in the probability of detecting the photon, which is the complex conjugate squared of its wavefunction. The superposition of a great number of wavefunctions generates the classical wave where the electric and magnetic fields carry the energy of the wave. (please note that superposition is the addition of wavefunctions, not an interaction)
In this diagram of how a polarized beam is built up by the spin direction of the photon, you may develop an intuition of the matter. 


Left and right handed circular polarization, and their associate angular momenta.

The red arrow is the direction of the maximum of the electric field ( the magnetic can be considered as perpendicular to it and to the direction of motion, and given)
The quantum field theory of how a classical field emerges  from the underlying quantum mechanical field ( as described by the answer of WetSavannaAnimal aka Rod Vance) can be seen in this blog entry.
In a very real sense the classical electromagnetic wave is riding on the photons.
