The imaging quality of optical systems is affected by aberrations caused by deviations between the actual optical path and the ideal optical path. Zernike polynomials, as an effective tool for describing aberrations, can describe and analyze the characteristics of optical aberrations. This paper reviews the research progress at home and abroad, and focuses on analyzing the mathematical expression forms of standard Zernike polynomials, Zernike circular polynomials, and Zernike annular polynomials, clarifying their corresponding relationships with typical aberrations such as spherical aberration, coma, and astigmatism. Research shows that Zernike circular polynomials can efficiently represent the distribution of axisymmetric aberrations through the characteristics of orthogonal basis functions in polar coordinates, while annular polynomials are suitable for describing off-axis aberrations.