Conduction is the transport of charge, and it doesn't matter what the charge carriers are. In metals, the charge carriers are, as you say, the conduction electrons. However, in electrolyte solutions like saline, the conductors are hydrated ions like Na+ and Cl−. As far as electricity is concerned our bodies are basically just dilute electrolytes, so the (rather low) conductivity of the human body is due to transport of ions. Many of the cells in your body are loaded with mobile ions like Calcium in order to perform their basic functions, and blood carries ions both to deliver them where they're needed and to maintain osmotic pressure and acidity. In addition, human blood contains elements that are called electrolytes that have electrical properties.