In this paper, we propose a relay selection amplify-and-forward (RS-AF) protocol in general bidirectional relay networks with two sources and N relays. In the proposed scheme, the two sources first simultaneously transmit to all the relays, and then, a single relay with a minimum sum symbol error rate (SER) will be selected to broadcast the received signals back to both sources. To facilitate the selection process, we propose a simple suboptimal min-max criterion for relay selection, where a single relay that minimizes the maximum SER of two source nodes will be selected. Simulation results show that the proposed min-max selection has almost the same performance as the optimal selection with lower complexity. We also present a simple asymptotic SER expression and make a comparison with the conventional all-participate AF relaying scheme. The analytical results are verified through simulations. To improve the system performance, optimal power allocation (OPA) between the sources and the relay is determined based on the asymptotic SER. Simulation results indicate that the proposed RS-AF scheme with OPA yields considerable performance improvement over an equal-power-allocation scheme, particularly with a large number of relay nodes.