What is causality? I was looking a few videos on youtube and I came across a video named the speed of light is nothing related to light. In the video, it was said that speed of light is actually speed of causality so what actually is causality?
 A: In the broadest sense, a world which exhibits causality is one in which 1) things have causes (i.e., magic does not operate) and 2) those causes precede their effects. As pointed out by PM 2Ring, a strict definition of this can be furnished in a space-time diagram, in terms of the past and future light-cones and their relationship to the world-line corresponding to you, the observer. 
Those light-cones delineate the locations in time and in space from which things in your past can influence something or someone in your present, and also whether or not things in the present can influence something or someone in the future- depending on where they are and when in the future. 
The boundaries between the spacetime zones where influence is possible and where it is not are set by the speed of light, which connects this concept of causality with c. The reason that "causality" travels "at the speed of light" is that no cause can have any effect which travels through spacetime faster than that.
A: Niels Nielsen is right. I am adding an answer which I hope will read more simply.
There is a maximum speed at which one physical thing can influence another physical thing located some distance away. Let us call that speed "the maximum speed for signals". The word "signal" is here referring to any sort of physical influence that could change what goes on at the receiving end.
Let's call this maximum speed $v_{\rm max}$.
Let's use $c$ for the speed of light in vacuum.
It is found that $c = v_{\rm max}$. In other words, electromagnetic waves in vacuum travel at the maximum speed for signals. For this reason, the maximum speed for signals is often referred to as "the speed of light". But if it so happened that electromagnetic waves traveled at some smaller speed, this would not affect the role and significance of $v_{\rm max}$. All the structure and ideas of special relativity would remain unchanged. For example, effects such as time dilation and Lorentz contraction do not involve any direct connection to light and electromagnetism. 
If, say owing to some sort of quantum structure throughout spacetime, it turned out that light really goes a little slower than $v_{\rm max}$ then very likely physicists would continue to use $c$ for the maximum speed for signals, and introduce some other letter to refer to the speed of light. But the experimental evidence is that light travels at $v_{\rm max}$ to high precision (it is measured using astronomical observations about the timing of the reception of X rays from binary stars, and things like that).
