Utilizing intelligent reflecting surface (IRS) was proven to be efficient in improving the energy efficiency for wireless networks. In this paper, we investigate the passive beamforming and channel estimation for IRS assisted wireless communications with low-resolution analog-to-digital converters (ADCs) at the receiver. We derive the approximate achievable rate by using the Bussgang theorem. Based on the derived analytical achievable rate expression, we maximize the achievable rate by using semidefinite programming (SDP), branch-and-bound (BB), and gradient-based approaches. A maximum likelihood (ML) estimator is then proposed for channel estimation by considering the $\mathrm{1}$-bit quantization ADC. Numerical result shows that the proposed beamforming design and channel estimation method significantly outperforms the existing methods.