Wat is Arrow–Debreu security and state price?

The Arrow–Debreu security of state $s$ pays 1 if state $s$ occurs and nothing otherwise. Its price $q_s$ is the state price of $s$. A vector $q\in\R^S$ is a state-price vector for the market if it prices every traded asset: $p = D^\top q$, that is $p_j=\sum_s q_sD_{sj}$.