The paper provides a comprehensive tutorial on intelligent reflecting surface (IRS)-aided wireless communications, addressing its key technical challenges from a communication perspective. IRS is a planar surface with passive reflecting elements that can dynamically adjust signal reflection to enhance wireless channel performance. The paper covers the fundamentals of IRS-aided wireless communication, including signal and channel models, hardware architecture, and practical constraints. It discusses the optimization of passive reflections, channel estimation, and deployment strategies for IRS in various system setups, such as single-user, multi-user, single-antenna, multi-antenna, narrow-band, broadband, and multi-cell communications. The paper also highlights the potential of IRS in achieving sustainable capacity growth in future wireless networks, despite the challenges of reflection optimization, channel estimation, and deployment. Additionally, it outlines future research directions, emphasizing the need for further investigation into mutual coupling effects, angle-dependent reflection, and frequency-dependent phase shifts.The paper provides a comprehensive tutorial on intelligent reflecting surface (IRS)-aided wireless communications, addressing its key technical challenges from a communication perspective. IRS is a planar surface with passive reflecting elements that can dynamically adjust signal reflection to enhance wireless channel performance. The paper covers the fundamentals of IRS-aided wireless communication, including signal and channel models, hardware architecture, and practical constraints. It discusses the optimization of passive reflections, channel estimation, and deployment strategies for IRS in various system setups, such as single-user, multi-user, single-antenna, multi-antenna, narrow-band, broadband, and multi-cell communications. The paper also highlights the potential of IRS in achieving sustainable capacity growth in future wireless networks, despite the challenges of reflection optimization, channel estimation, and deployment. Additionally, it outlines future research directions, emphasizing the need for further investigation into mutual coupling effects, angle-dependent reflection, and frequency-dependent phase shifts.