This paper studies the active intelligent reflecting surface (IRS)-enabled integrated sensing and communications (ISAC), in which an active IRS is deployed to assist the base station (BS) in serving multiple communication users (CUs) and simultaneously sensing an \emph{extended} target at the non-line-of-sight (NLoS) area of the BS. The active IRS has the capability of amplifying the reflected signals so as to overcome significant reflection path loss in NLoS communication and sensing. In particular, we derive the sensing Cramér-Rao bound (CRB) for estimating the target response matrix. Accordingly, we jointly optimize the transmit beamforming at the BS and the reflective beamforming at the active IRS to minimize the sensing CRB, subject to the signal-to-interference-plus-noise ratio (SINR) requirements at the CUs, the transmit power budgets at the BS and active IRS, as well as the power amplification gain constraints at the active IRS. The CRB minimization problem is highly non-convex and thus difficult to solve in general. To address this challenge, we first focus on two specified conditions by considering the sensing-only scenario via ignoring the SINR constraints for communications, for which the closed-form optimal transmit beamforming is derived. Then, we propose two efficient alternating optimization (AO)-based algorithms to obtain high-quality solutions for the general ISAC scenarios. Next, we analyze the inherent relationship between the power scaling at the BS and the amplification scaling at the active IRS. It is shown that the active IRS always amplifies the signal using the maximum amplification gain under practical system settings. Finally, numerical results are provided to verify the effectiveness of the proposed AO-based algorithms and the benefits of active IRS-enabled ISAC compared to its passive IRSs counterparts.