What organ system does the kidney belong to? What organ system does the liver belong to?

Kidneys are part of the excretory system (urinary system). This system is associated with excretion of metabolic waste products and formation and excretion of urine. Other organs of the system are ureters, urinary bladder and urethra.

The diagram of excretory system :

Liver is a part of the alimentary system (other name is digestive system). This system has two main parts : Digestive tract and digestive glands. From mouth to anus all organs (mouth, esophagus, stomach, small intestine, large intestine, rectum and anus) form a tract, which is called digestive tract.
Salivary glands, Liver and pancreas are part of digestive glands.

As the name suggests the digestive system is associated with digestion, absorption and metabolism of nutrients. Liver forms bile, which is essential for fat digestion and absorption. And liver is the organ of metabolism. So it belongs to alimentary (digestive) system.

A Diagram of alimentary (digestive) system :

Sometimes Liver, gall bladder and bile ducts (the organs related to bile production and secretion) are included into a system called the Hepatobiliary system.