What does $\frac{d^2 u}{dt^2}$ mean? When it comes to taking a derivative, what does $\displaystyle \frac{d^2 u}{dt^2}$ mean ? Does it mean taking derivative of the function twice with respect to $t$. If yes, why is then $d^2 u$ squared? Thanks in advance!
This basically means that taking the second derivative of any given function.
To elaborate this you can consider some function $$y=f(t)$$ Now lets find its derivative $$y'= \frac{d(f(t))}{dt}$$. Now lets say I want to know what would the derivative of the earlier function $y'$ would be. That will be $$y''=\frac{d(\frac{d(f(t))}{dt})}{dt}$$ and this would be $$y''=\frac{d^2u}{dt^2}$$ where $u=f(t)$ 

Now in terms of physics you can think this as if i have the position as the function of time if i find its derivative it gives me the velocity and again if i find its derivative it gives me the acceleration . So the second derivative of the position gives me the acceleration
this simply means it is the second derivative of u with respect to t