Definition
The quadratic variation of a process $(Z_t)$ is
$$ \langle Z\rangle_t
\lim_{N\to\infty}\sum_{i=0}^{N-1}(Z_{t_{i+1}}-Z_{t_i})^2. $$
Brownian motion case
For Brownian motion,
$$ \langle B\rangle_t=t. $$In differential form,
$$ d\langle B\rangle_t=dt. $$Why it matters
This identity explains why the second-order term in Ito formula turns into a $dt$ term instead of vanishing.