A series of Binarized Neural Networks (BNNs) show the accepted accuracy in image classification tasks and achieve the excellent performance on field programmable gate array (FPGA). Nevertheless, we observe existing designs of BNNs are quite time-consuming in change of the target BNN and acceleration of a new BNN. Therefore, this paper presents FCA-BNN, a flexible and configurable accelerator, which employs the layer-level configurable technique to execute seamlessly each layer of target BNN. Initially, to save resource and improve energy efficiency, the hardware-oriented optimal formulas are introduced to design energy-efficient computing array for different sizes of padded-convolution and fully-connected layers. Moreover, to accelerate the target BNNs efficiently, we exploit the analytical model to explore the optimal design parameters for FCA-BNN. Finally, our proposed mapping flow changes the target network by entering order, and accelerates a new network by compiling and loading corresponding instructions, while without loading and generating bitstream. The evaluations on three major structures of BNNs show the differences between inference accuracy of FCA-BNN and that of GPU are just 0.07%, 0.31% and 0.4% for LFC, VGG-like and Cifar-10 AlexNet. Furthermore, our energy-efficiency results achieve the results of existing customized FPGA accelerators by 0.8× for LFC and 2.6× for VGG-like. For Cifar-10 AlexNet, FCA-BNN achieves 188.2× and 60.6× better than CPU and GPU in energy efficiency, respectively. To the best of our knowledge, FCA-BNN is the most efficient design for change of the target BNN and acceleration of a new BNN, while keeps the competitive performance.