The act of giving birth (parturition) in animals is a very involved process involving several physiological processes controlled by hormones. The answer is beyond the limited space available for this answer. However, we can provide you with several links where you can read more about how animals give birth.
• Horse Behavior at Parturition
• Beef Cattle Parturition
• Parturition in Horses
• Signs of Impending Calving in Cows or Heifers
• Mare Reproduction
• The Beef Cow Estrous Cycle
• Assisting Cows at Calving