This paper examines the information theoretic metric of outage probability for a decode-and-forward (DF) based asymmetric two-way relay transmission (i.e. the two source terminals have different target rates). We first characterize the achievable rate region of a conventional three-node network. After that the conventional three-node scenario is analyzed in terms of outage probability and the corresponding closed-from expressions are developed over Rayleigh fading channels. Moreover, in order to make a good use of the available diversity degrees of the channel, opportunistic relay selection are considered for multi-relay networks. Two significant relay selection strategies, i.e., the max-min and max-sum policies are studied and analyzed in terms of outage probability and diversity gain from the viewpoint of asymmetric traffics. Furthermore, a single-criterion based relay selection policy is proposed, which only uses the harmonic mean of the two-hop squared link strengths, thus in contrast to the hybrid scheme no additional overhead is required during the relay selection process. Numerical experiments are done and outage performance comparisons are conducted. Our results show that the proposed policy is an efficient and appropriate method to implement relay selection and can achieve significant performance gains in terms of outage probability regardless of the symmetry and asymmetry of the traffics and channels. Moreover, the simulation results also validate the accuracy of our derived expressions.