The obturator nerve is one of several motor nerves that innervates the powerful muscles of the pelvic limb. In dogs, it arises from the caudal lumbar spinal nerves, specifically L4-L6. It dives in to the iliopsoas muscle caudomedially and exits in a dorsomedial fashion along the belly of the muscle. From here, it travels along the body of the ilium and enters the levator ani medially. It exits the pelvis via the cranial portion of the obturator foramen, innervating the external obturator muscle in the process. It continues on to innervate the gracilis, pectineus, and adductor muscles.
Because this nerve innervates the adductors of the hip, lesions result in an inability to adduct the limb, causing splayed hind limbs. This would be most easily seen as a problem walking on slick surfaces-- difficult to do when you can't get your legs back to the mid-line of the body. Loss of innervation to the external obturator causes a loss of the muscle's ability to rotate the limb laterally. This is, however, of little consequence, as the internal obturator muscle performs the same activity and is innervated instead by the sciatic nerve. The obturator nerve may be damaged during parturition, or possibly due to trauma.
On a dissection specimen, the obturator is most easily seen as it exits the iliopsoas, or as it enters the adductor.