What organ system is responsible for breaking down food into molecules that can be absorbed by the body?

The system you're looking for should be "digestive system".

This organ system consists of organs such as mouth, stomach, intestines etc.

Basically, how this system works is that it breaks down complex and insoluble food molecules by using enzymes (chemical digestion) , into simple and soluble molecules which can be absorbed by the body and be turned to energy for our body to work. Examples of chemical digestion is such as enzyme lipase digesting lipids into fatty acids and glycerol. In addition, physical digestion (without enzymes) is also used to help speed up chemical digestions. Examples of physical digestion is where we chew the food we eat into a bolus, which can increase the surface area of the food and the enzyme digestion later on can be faster and more efficient.