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In this paper we study an intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) network where the direct link between the base station (BS) and one of the users is blocked and the IRS is deployed to serve the blocked user. The IRS designs under both the ideal IRS with continuous phase shifting and the non-ideal IRS with discrete phase shifting are considered. For both cases, by leveraging the isotropic random vector and the Laguerre series, we derive insightful results and closed-form expressions on the performance measures including the average required transmit power, the outage probability, and the diversity order. Our analytical results show that the transmit power scales down linearly with the BS antenna number and quadratically with the IRS element number. The diversity order equals the smaller of the BS antenna number and the IRS element number with a scaling coefficient. Our results also reveal the effect of the phase quantization resolution to the system performance when non-ideal IRS is used. Numerical results are provided to validate the accuracy of our analysis and the non-ideal IRS with four or more bits for quantization is shown to achieve nearly the same performance as the ideal IRS.