Context. The Sun is very quiet with less sunspots and activity since the beginning of solar cycle 24. However, the active region (AR) 11 045 emerged on February 5, 2010, is associated with 43 (8 M-and 35 C-class) flares, 53 coronal mass ejections (CMEs), 29 filament eruptions, 19 extreme ultraviolet (EUV) waves and abundant jets, indicating that this AR is the first productive one of solar cycle 24. Aims. We study the AR evolution and its associated activities, and also their relationships, to understand this productive AR. Methods. We used SOHO/MDI magnetograms to study the magnetic fields, STEREO/SECCHI images to explore the activities, and GOES measurements to investigate the soft X-ray flux of the AR. Results. During the AR evolution, six pairs of main magnetic fields emerged, and 93.1% flares and 82.75% filament eruption occurred in the emergence and stable phases of the magnetic flux. However, 43.4% CMEs occurred in the decaying phase, even though there were less flares. An example is given to show that an event is related to a flare, a filament eruption, a CME and an EUV wave from inner corona to outer corona in space, and the filament eruption and EUV wave occur near the peak time of the flare. Among the 29 filament eruptions, 79.3% are associated with CMEs, as well as 58.6%, associated with flares, and 34.5%, associated with EUV waves. During the 12-day active phase, 575 jets are detected with a daily occurrence rate of 49.3. This is the first time that so many jets have been identified in one AR, implying at least 575 lower magnetic reconnection processes during the AR evolution. We statistically studied these jets along with the AR evolution, and noticed that the jets mostly occurred surrounding the emerging flux. We also investigated the spatio-temporal relationships between the jets and the flares, and find that the jets are usually rooted around the flare cores, and the soft X-ray flux is inverse correlated with the number of the jets, especially during the beginning 9 days since the AR emergence. In comparison with AR 11045, we studied the other newly emerging AR 11045, and obtained similar results. The relationships between the jets and the flares may well represent a scenario of two-step magnetic reconnection. Using schematic diagrams, we explain the remarkable magnetic field emergence, cancelation and shear motion of AR 11045, and its associated activities.