Ito's formula, and the relationship between dt and dB(t) reading Bernt Oksendal Stochastic Differential Equations. 
I have just seen Ito's Formula, after this the author then says where $dX(t)^{2}$ is calculated using
 $$dt\cdot dt=dt \cdot dB(t) =dB(t)\cdot dt =0$$
 and
 $$dB(t) \cdot dB(t) = dt$$
On page 45 at the top. Where do these relations come from, what is the intuition behind it. 

$dt\cdot dt=0$ is shorthand notation for the fact that the quadratic variation of the identity function with itself is $0$.
$dt\cdot dB(t)=0$ is shorthand notation for the fact that the cross variation of Brownian motion with the identity function is $0$.
$dB(t)\cdot dB(t)=dt$ is shorthand notation for the fact that the quadratic variation of Brownian motion with itself over a time interval of length $t$ is equal to $t$.
