According to the Brønsted-Lowry theory, an acid is a proton donor and a base is a proton acceptor. In this reaction, $NH_3$ accepts a proton from $H_2O$ to become $NH_4^+$, so $NH_3$ is the base. $H_2O$ donates a proton to become $OH^-$, so $H_2O$ is the acid.