This paper studies relay selection and power allocation for amplify-and-forward (AF) based two-way relay networks (TWRN) with asymmetric traffic requirements (ATR). A joint relay selection and power allocation algorithm is proposed to decrease the outage probability of TWRN with ATR. In this algorithm, two sources exchange information with the help of the relay during two time slots. We first calculate the optimal power allocation parameters based on instantaneous channel state information (CSI), and then derive a tight lower bound of outage probability. Furthermore, we propose a simplified relay selection criterion, which can be easily calculated as harmonic mean of instantaneous channel gains, according to the outage probability expressions. Simulation results verified the theoretical analyses we presented. It is shown that the outage probability of our algorithm improves 3-4dB comparing with that of other existing algorithms, and the lower bound is tight comparing with actual value for the entire signal-to-noise ratio (SNR) region.