In his article, Dupire (1994) developed the local volatility approach
under the assumption that options are traded for a continuum of maturities
and strikes. In reality, only a finite number of options generating a grid of strikes and
maturities is traded. Then the reconstruction of the local volatility function is
obtained by interpolation methods.
Yes looks like ATM volatility. It’s forward (he also calls it forward forward volatility). Say you have the volatility of an option with 30 days maturity, $\sigma_1$ and $T_1$; and the volatility of an option with 60 days maturity, $\sigma_2$ and $T_2$.The 0-30 bucket will have $\sigma_1$ , whereas the 30-60 days bucket will have the forward volatility ...