Intelligent reflecting surface (IRS) has garnered growing interest and attention due to its potential for facilitating and supporting wireless communications and sensing. This paper studies a semi-passive IRS-enabled sensing system, where an IRS consists of both passive reflecting elements and active sensors. Our goal is to minimize the Cramér-Rao bound (CRB) for parameter estimation under both point and extended target cases. Towards this goal, we begin by deriving the CRB for the direction-of-arrival (DoA) estimation in closed-form and then theoretically analyze the IRS reflecting elements and sensors allocation design based on the CRB under the point target case with a single-antenna base station (BS). To efficiently solve the corresponding optimization problem for the case with a multi-antenna BS, we propose an efficient algorithm by jointly optimizing the IRS phase shifts and the BS beamformers. Under the extended target case, the CRB for the target response matrix (TRM) estimation is minimized via the optimization of the BS transmit beamformers. Moreover, we explore the influence of various system parameters on the CRB and compare these effects to those observed under the point target case. Simulation results show the effectiveness of the semi-passive IRS and our proposed beamforming design for improving the performance of the sensing system.