Integrating BD-RIS into wireless communications systems has attracted significant interest due to its transformative potential in enhancing system performance. This survey provides a comprehensive analysis of BD-RIS technology, examining its modeling, structural characteristics, and network integration while highlighting its advantages over traditional diagonal RIS. Specifically, we review various BD-RIS modeling approaches, including multiport network theory, graph theory, and matrix theory, and emphasize their application in diverse wireless scenarios. The survey also covers BD-RIS's structural diversity, including different scattering matrix types, transmission modes, intercell architectures, and circuit topologies, showing their flexibility in improving network performance. We delve into the potential applications of BD-RIS, such as enhancing wireless coverage, improving PLS, enabling multi-cell interference cancellation, improving precise sensing and localization, and optimizing channel manipulation. Further, we explore BD-RIS architectural development, providing insights into new configurations focusing on channel estimation, optimization, performance analysis, and circuit complexity perspectives. Additionally, we investigate the integration of BD-RIS with emerging wireless technologies, such as millimeter-wave and terahertz communications, integrated sensing and communications, mobile edge computing, and other cutting-edge technologies. These integrations are pivotal in advancing the capabilities and efficiency of future wireless networks. Finally, the survey identifies key challenges, including channel state information estimation, interference modeling, and phase-shift designs, and outlines future research directions. The survey aims to provide valuable insights into BD-RIS's potential in shaping the future of wireless communications systems.