What exactly is the string theory? Do these 'strings' really exist? And how can we know they exist if they have not been proven?

String theory is essentially trying to explain why we have all the fundamental particles we have.  So, protons and neutrons are made up of even smaller particles called quarks, and you get electrons and higher energy, more exotic particles – things like muons, and all sorts of different particles.  They can combine in different ways and they all have different masses.

People who are much better at maths  have spent a long time trying to find mathematical constructs, so ways of putting maths together to produce objects which look like the particles that we see and have similar properties which we can call mass. 

One of the ways they've done this is with some maths which look a bit like strings.  You can have oscillations on a string.  If you wobble the string slowly, you get one wobble in it as it wobbles left and right.  If you wobble it faster, you can get it to start making a snaking wobble.  As you make it faster and faster, these different vibrations could be associated with different particles.

The actual strings themselves are probably just maths.  We have no evidence to say there are actual little bits of cotton wobbling very, very rapidly.  So all we know is that there is some maths which gives rise to things which look a bit like the particles we have.  We are not even sure that there's any actual evidence to say that string theory is better than any other particular theory.  They haven’t actually got that far, but that's what the guys in CERN are trying to do.