Special theory of relativity is basically this:
Every observer has their own inertial frame - perspective on the universe (this is to do with them having a particular velocity)
In your inertial frame, time and space warp such that the speed of light always appears to be constant.
So if you go faster, time dilates (each second becomes "longer") and lengths contract (metres become "shorter"). Without diagrams, it's quite hard to explain why this would make the speed of light look like it is constant regardless of your personal velocity, so maybe look in a physics textbook or check out t'internet.
As for theories that dispute the fact that the speed of light is constant, the only type I know are Variable Speed of Light (VSL) theories. These theories don't dispute relativity out of hand i.e. they still agree that right now, if I was going faster than you, time would pass more slowly for me etc.... What they suggest is that in the early universe, the speed of light was still constant for all observers, but was different to what it is today i.e. they are challenging the numerical value of the speed of light at different times in the universe's history.
Joao Magueijo's book Faster Than the Speed of Light is about VSL and its implications for cosmology. Also has a really good introduction to special and general relativity. Well worth reading if you're doing your coursework on this.