The Orion Nebula (NGC 1976, M 42) is an H II region composed of a slowly expanding thin zone of photoionized gas on the facing side of the Orion Molecular Cloud. The Orion Nebula is also a famous star formation region in which numerous jets and shocks arise from many young stars. Creating bipolar jets to shed excess angular momentum is an essential process in star formation. The jets interact with the interstellar medium or with wakes of previously passing jets and subsequently form shocks. These shocks can be observed with optical or near infrared emission lines and are called Herbig-Haro (HH) objects. The purpose of the present study was to catalog and study the HH objects in the Orion Nebula, and hence, this will help us understand how stars form in a molecular cloud and then evolve in an H II region. We measured the proper motions (tangential velocities) and radial velocities of the HH objects with the highest possible accuracy. By combining the results of the proper-motion and radial velocity measurements, we could obtain the spatial (three-dimensional) motions of the HH objects, which gave us the opportunity to discover the true physical nature of the HH objects in the Orion Nebula. We were able to measure the proper motions of the HH objects with a 10 km s-1 accuracy using the Hubble Space Telescope (HST) Wide Field Planetary Camera 2 (WFPC2) images in [S II], [N II], Halpha, and [O III], taken 4 to 6 years apart. This is the first study dedicated to measuring the proper motions of HH objects in the Orion Nebula covering the complete range of ionization states. A shock consists of a shock front followed by a collisional excitation zone and a cooling zone in which [O III], [N II], and [S II] emission layers form. Halpha emission comes directly from the collisional excitation zone. The presence of [O III], [N II], or [S II] emissions is a good indicator of the strength of a shock. We identified all the HH flows in the northwest and southeast regions of the Orion Nebula and found two new bipolar objects. We confirmed that the major HH flows in the northwest region were created by an explosive event which took place approximately 1000 years ago. We also found a new HH flow moving toward the northeast in the southwest region of the Orion Nebula. The proper motion of HH 202 was measured with high accuracy for the first time, and the result indicates that HH 202 and HH 203/204 may have emanated from the same unknown origin. (Abstract shortened by UMI.)