The bidirectional relay channel, in which two users communicate with each other through a relay node, is a simple but fundamental and practical network architecture. In this paper, we consider the block fading bidirectional relay channel with a decode-and-forward relay and propose efficient transmission strategies that exploit the block fading property of the channel. We assume that a direct link between the two users is not present and consider two transmission modes: 1) the multiple-access mode (both users transmit to the relay) and 2) the broadcast mode (the relay transmits to both users). Most existing relaying protocols assume a fixed schedule for using these transmission modes. In contrast, we abandon the restriction of having a fixed and predefined schedule and propose to optimize the selection of the transmission modes and the associated transmission rates based on the instantaneous channel state information (CSI) of the involved links. Thereby, we consider two different types of transmit power constraints: 1) a fixed transmit power for each node and 2) a per-node long-term power constraint. To enable the use of a nonpredefined schedule for transmission mode selection, the relay has to be equipped with two buffers for storage of the information received from both users. We develop new relaying protocols based on adaptive mode selection and provide the corresponding achievable long-term rate regions. In particular, based on the CSI of the involved links, the optimal transmission mode as well as the optimal transmission rates and/or the transmit powers of the nodes are chosen in each time slot to maximize the weighted sum rate of both users. By varying the weights assigned to the users, the boundary surface of the achievable long-term rate region of the proposed protocol can be obtained. In addition, we discuss and address two practical challenges for the implementation of the proposed protocols, namely, the availability of the knowledge of the channel statistics required for the implementation of the optimal protocols, and the increase of the end-to-end delay due to the data buffering. Numerical results confirm the superiority of the proposed buffer-aided protocols compared with existing bidirectional relaying protocols.